Final Report on the 

Evaluation of the Growth Model Pilot Project

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Final Report on the Evaluation of the 

Growth Model Pilot Project

Prepared by:

Thomas B. Hoffer

E. C. Hedberg

Kevin L. Brown

Marie L. Halverson

Paki Reid-Brossard

National Opinion Research Center (NORC)

at the University of Chicago

Andrew D. Ho, Harvard Graduate School of Education

Katherine Furgol, University of Iowa

Prepared for:

U.S. Department of Education

Office of Planning, Evaluation and Policy Development

Policy and Program Studies Service

2011
This report was prepared for the U.S. Department of Education under Contract Number ED-06-

CO-0053 with National Opinion Research Center at the University of Chicago. Andrew Abrams 

served as the contracting officer’s representative. The views expressed herein do not necessarily 

represent the positions or policies of the Department of Education. No official endorsement by the 

U.S. Department of Education is intended or should be inferred.

U.S. Department of Education

Arne Duncan

Secretary

Office of Planning, Evaluation and Policy Development

Carmel Martin

Assistant Secretary

Policy and Program Studies Service

Adriana de Kanter

Acting Director

January 2011

This report is in the public domain. Authorization to reproduce it in whole or in part is granted. While 

permission to reprint this publication is not necessary, the suggested citation is: U.S. Department of 

Education, Office of Planning, Evaluation and Policy Development, Policy and Program Studies Service, 

Final Report on the Evaluation of the Growth Model Pilot Project, Washington, D.C., 2011.

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On request, this publication is available in alternate formats, such as Braille, large print or computer 

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or 202-260-0818.
Evaluation of the Growth Model Pilot Project iii

Contents

List of Exhibits.............................................................................................................................. v

Acknowledgments........................................................................................................................ xi

Executive Summary...................................................................................................................xiii

GMPP Objectives .................................................................................................................xiii

States Included in the Pilot Project....................................................................................... xiv

Findings ................................................................................................................................ xiv

Features of Growth Models Implemented by Pilot States............................................. xiv

Comparing AYP by Growth with Status and Safe-Harbor............................................. xv

Impact of Growth Models on AYP Rate Among High-Poverty Schools....................... xvii

Hypothetical Results of Using Growth Models Instead of Status and 

Safe-Harbor.................................................................................................................xviii

Effects of Growth Model Types on Student and School Outcomes ............................... xix

Predictive Accuracy of the Different Types of Growth Models...................................... xx

Effects of Alternative Standards of Adequate Yearly Growth ........................................ xx

Qualifications and Implications............................................................................................ xxi

I. Introduction............................................................................................................................ 1

Scope of This Report ............................................................................................................... 1

The Status Model of Accountability Under ESEA .................................................................. 3

Limitations of the Status Model .............................................................................................. 5

The Growth Model Pilot Project (GMPP)............................................................................... 5

Types of Growth Models Implemented in the Pilot .............................................................. 10

How Growth Model Results Are Used for School AYP Determinations ............................. 14

Data Sources and Availability ............................................................................................... 19

Calculation of Hypothetical “Growth-Only” Outcomes................................................ 20

Data Limitations .................................................................................................................... 20

II. Effects of Growth Models Implemented Under the GMPP on School AYP 

Determinations..................................................................................................................... 23

State Growth Models and Their Effects on Schoolwide AYP Results.................................. 23

Alaska’s Growth Model.................................................................................................. 23

Arizona’s Growth Model ................................................................................................ 25

Arkansas’ Growth Model ............................................................................................... 27

Delaware’s Growth Model............................................................................................. 28

Florida’s Growth Model................................................................................................. 30

Iowa’s Growth Model..................................................................................................... 31

North Carolina’s Growth Model.................................................................................... 32

Ohio’s Growth Model..................................................................................................... 33

Tennessee’s Growth Model ............................................................................................ 35

Impact of GMPP on AYP...................................................................................................... 36

Impact of the GMPP on Subgroup AYP Outcomes .............................................................. 37

Discussion.............................................................................................................................. 40
Evaluation of the Growth Model Pilot Project iv

III. School Characteristics Associated with AYP Outcomes.................................................. 43

Discussion.............................................................................................................................. 47

IV. Characteristics of Growth Models Affecting AYP Outcomes......................................... 49

Comparison of Student On-Track and Proficiency Results................................................... 49

Effects of Order of Application of Growth, Status, and Safe-Harbor ................................... 52

Effects of Types of Models.................................................................................................... 56

Generalizing From a Standardized North Carolina Dataset......................................... 58

Generic Growth Models................................................................................................. 59

The Generic Transition Matrix Model .................................................................... 59

The Generic Trajectory Model................................................................................ 60

The Generic Projection Model................................................................................ 61

Generic Model Extensions for Growth-Only Results.............................................. 62

Student-Level Results for Status-Plus-Growth Models .................................................. 63

Student-Level Results for Growth-Only Models............................................................. 68

Alternative Cut Scores for Status-Plus-Growth Models................................................. 71

A Framework for Comparing the Models ...................................................................... 73

Extending the Framework to the Growth-Only Results ................................................. 79

School AYP Simulations Based on the Generic Models................................................. 81

Consistency of AYP Determinations With Future Student Learning.............................. 83

Comparison of Generic Growth Models: Summary and Conclusions........................... 87

Effects of Different Growth Standards for School Accountability ....................................... 88

Effects of Longitudinal Matching.......................................................................................... 91

Conclusions............................................................................................................................ 92

Conclusions.................................................................................................................................. 95

How have states in the pilot project implemented growth models? ...................................... 95

How did each pilot state’s growth model affect the number and kinds of schools 

that make AYP?..................................................................................................................... 96

Implications for Future Policy............................................................................................... 98

References.................................................................................................................................. 103

Appendix A: Comparison of GMPP Growth Models with State Accountability

Systems............................................................................................................... 105

Appendix B: State GMPP Model Summaries........................................................................ 111

Alaska ........................................................................................................................... 111

Arizona ......................................................................................................................... 113

Arkansas....................................................................................................................... 115

Delaware ...................................................................................................................... 118

Florida.......................................................................................................................... 122

North Carolina ............................................................................................................. 124

Tennessee...................................................................................................................... 127

Appendix C: Supplemental Exhibits....................................................................................... 129

Appendix D: Derivation of the Generic Projection Model Rule for Identifying 

On-Track Students............................................................................................ 137
Evaluation of the Growth Model Pilot Project v

Exhibits

Exhibit ES.1 Percentage of Schools That Made AYP Before and After the Application 

of the Growth Model, in Nine States, in 2007–08 and 2006–07 ......................... xvi

Exhibit ES.2 Number of Schools Making or Not Making AYP By Status, Safe-Harbor 

or Growth Model, Percentage Increase in Number of Schools That Made 

AYP Due to Growth, and Percentage Decrease in Number of Schools 

That Did Not Make AYP Due to Growth, by State, 2006–07 ............................ xvii

Exhibit 1 Determining AYP Under the Status Model ............................................................ 4

Exhibit 2 Relationship of Growth and Status Models for Assessing Achievement 

Proficiency .............................................................................................................. 6

Exhibit 3 Seven Core Principles of the Growth Model Pilot Project ..................................... 7

Exhibit 4 Determining AYP Under Status, Safe-Harbor, and Growth................................... 9

Exhibit 5 Illustration of How the Iowa Transition Matrix Model Classifies Students......... 10

Exhibit 6 Illustration of How the Delaware Transition Matrix Model Assigns Points 

to Students’ Gains................................................................................................. 11

Exhibit 7 Illustration of How Trajectory Models Set Targets for Students Scoring 

Below Proficiency Thresholds.............................................................................. 12

Exhibit 8 Illustration of How a Projection Model Predicts Future Achievement 

Based on Prior Scores........................................................................................... 14

Exhibit 9 Overview of Growth Models Approved for the GMPP in the 2007–08 

School Year........................................................................................................... 17

Exhibit 10 Alaska School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08........... 25

Exhibit 11 Arizona School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08.......... 27

Exhibit 12 Arkansas School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08.......... 28

Exhibit 13 Delaware School AYP Determinations With Growth Model Results 

Augmented With Status and Safe-Harbor Results, 2006–07 and 2007–08.......... 29

Exhibit 14 Delaware School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08........... 30

Exhibit 15 Florida School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08........... 31

Exhibit 16 Iowa School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08........................ 32

Exhibit 17 North Carolina School AYP Determinations With Status and 

Safe-Harbor Results Augmented With Growth Model Results, 2006–07 

and 2007–08.......................................................................................................... 33
Evaluation of the Growth Model Pilot Project vi

Exhibit 18 Ohio School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2007–08 ............................................. 35

Exhibit 19 Tennessee School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08........... 36

Exhibit 20 Percentage Increase in Number of Schools That Made AYP Due to 

Growth, and Percentage Decrease in Number of Schools That Did Not 

Make AYP Due to Growth, by State, 2007–08 .................................................... 37

Exhibit 21 Number of Eligible Schools and Percentage in Which Reading AMO 

Was Met Because of Growth Model Results, by All Students and Racial 

or Ethnic Reporting Groups, 2007–08.................................................................. 38

Exhibit 22 Number of Eligible Schools and Percentage in Which Reading AMO 

Was Met Because of Growth Model Results, by Low-SES, SWD, and 

LEP Reporting Groups, 2007–08.......................................................................... 40

Exhibit 23 Numbers of Schools Making AYP by Status or Safe-Harbor, and 

Percentage Increase in Schools Making AYP Due to Growth, by NCLB

School Improvement Status, 2007–08 .................................................................. 44

Exhibit 24 Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and 

Percentage Increase in AYP Due to Growth, by School Poverty 

Concentration, 2007–08........................................................................................ 45

Exhibit 25 Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and 

Percentage Increase in AYP Due to Growth, by School Minority 

Concentration, 2007–08........................................................................................ 46

Exhibit 26 Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and 

Percentage Increase in AYP Due to Growth, by School Urbanicity, 

2007–08................................................................................................................. 47

Exhibit 27 Summary of the Effect of GMPP by School Demographic Characteristic 

and by State, 2006–07 and 2007–08..................................................................... 48

Exhibit 28 Conceptual Map of How Growth Model On-Track to Proficiency 

Designations Compare With Status Model Proficiency Designations for 

Students................................................................................................................. 50

Exhibit 29 Distribution of Students According to How Their Proficiency and 

On-Track to Proficiency Classifications Compare, 2007–08 .............................. 51

Exhibit 30 How the Growth Model On-Track-to-Proficiency Designations Can 

Compare With AYP Designations for Schools..................................................... 53

Exhibit 31 Percentage of Schools Meeting AMO Using Only the Growth Model 

On-Track Indicator, by Standard EDFacts AYP Classification and State, 

2007–08................................................................................................................. 54

Exhibit 32 Percentage of Schools Meeting the AMO Using Growth-Only and 

Making AYP by Other Means After Growth-Only Is Applied, by State, 

2007–08................................................................................................................. 56

Exhibit 33 Summary of Research Questions About the Generic Types of Growth 

Models................................................................................................................... 57

Exhibit 34 Illustration of Multiple Cut Scores for a Transition Matrix Model ...................... 58
Evaluation of the Growth Model Pilot Project vii

Exhibit 35 Student Weights Under the Generic Transition Matrix Model............................. 60

Exhibit 36 Schematic Diagram of Data Required for Applying the Projection 

Model.................................................................................................................... 62

Exhibit 37 Overall Proficiency Rates for Reading and Mathematics for Each Grade 

in 2006 .................................................................................................................. 64

Exhibit 38 “On-Track” Classification Rates for All Students for Reading and 

Mathematics by Model and Grade in 2006.......................................................... 65

Exhibit 39 “On-Track” Classification Rates for Eligible (Non-Proficient) Students 

for Reading and Mathematics by Model and Grade in 2006................................ 66

Exhibit 40 Pairwise Percent Agreement of Models for Eligible (Non-Proficient) 

Students, Reading and Mathematics..................................................................... 68

Exhibit 41 “On-Track” Classification Rates Using Growth-Only Results for All 

Students for Reading and Mathematics, by Model and Grade in 2006 ................ 69

Exhibit 42 Pairwise Percent Agreement of Models Using Growth-Only Results for 

All Students in Reading and Mathematics in 2006............................................... 70

Exhibit 43 “On-Track” Classification Rates for All Students for Reading and 

Mathematics by Model and Grade in 2006: Higher Cut Scores, 

Proficiency Rates Near 60 Percent ....................................................................... 72

Exhibit 44 “On-Track” Classification Rates for Eligible (Non-Proficient) Students 

for Reading and Mathematics by Model and Grade in 2006: Higher Cut 

Scores, Proficiency Rates Near 60 Percent........................................................... 73

Exhibit 45 A Bivariate Framework Showing the Non-Proficient Students Deemed 

“On-Track” by a Transition Matrix Model........................................................... 75

Exhibit 46 Areas of Discrepancy Between Transition Matrix and Trajectory Models.......... 77

Exhibit 47 The Projection Model Classification Approach.................................................... 78

Exhibit 48 Transition Matrix Model Classification Areas for On-Track Students 

in 2006 .................................................................................................................. 80

Exhibit 49 Areas Where Trajectory Models and Projection Models Will Disagree .............. 81

Exhibit 50 Percentage of Schools Making AYP by Status and by Growth Based 

on Results from the Different Types of Generic Growth Models ....................... 82

Exhibit 51 Percentage of Schools Making AYP by Growth-Only and 

Growth-Plus-Status Based on Results from the Different Types 

of Generic Growth Models. .................................................................................. 83

Exhibit 52 Correct Classification Rates by Model and Grade, Reading and 

Mathematics.......................................................................................................... 85

Exhibit 53 Percentage of On-Track-Classified Students Who Achieve Proficiency 

by Model and Grade, Reading and Mathematics.................................................. 86

Exhibit 54 Percentages of Students Scoring At or Above Proficiency, Making 

GMPP Annual Growth Targets (Upper Panel), Making Alternative 

Annual Growth Standard (Lower Panel), and Meeting Either Proficiency 

or Growth Target, Florida Data from 2006–07 and 2007–08............................... 89
Evaluation of the Growth Model Pilot Project viii

Exhibit 55 Average One-Year Gains in Mathematics and Reading, by Base Year 

Achievement Decile.............................................................................................. 90

Exhibit 56 Two-Year Match Rates for Students and Differences Between Matched 

and Unmatched Grade-Standardized Student Scores for Reading and 

Math, by State, 2007–08 ....................................................................................... 92

Exhibit A.1 Overview of Growth Components of State Accountability Systems.................. 106

Exhibit B.1 Illustration of Alaska’s Method for Determining Whether a Student Is 

On-Track Toward Proficiency ............................................................................ 112

Exhibit B.2 Arizona Proficiency Standards Test Cutoffs for Grades 3 Through 8................ 113

Exhibit B.3 Illustration of Arizona’s Method for Determining Whether a Student Is 

On-Track Toward Proficiency ............................................................................ 115

Exhibit B.4 Arkansas Benchmark Exam Proficiency Standards for Grades 3 

Through 8............................................................................................................ 116

Exhibit B.5 Arkansas Growth Target Multipliers for Grades 3 Through 7 ........................... 116

Exhibit B.6 Illustration of Arkansas’ Method for Determining Whether a Student 

Is On-Track Toward Proficiency ........................................................................ 118

Exhibit B.7 Example Comparison of Two Hypothetical Schools in Delaware to 

Illustrate Growth Model...................................................................................... 121

Exhibit B.8 Florida’s Cutoff Developmental Scale Scores to Be Considered Proficient 

on the FCAT for Grades 3 Through 7................................................................. 122

Exhibit B.9 Illustration of Florida’s Method for Determining Whether a Student Is 

On-Track Toward Proficiency in Reading.......................................................... 123

Exhibit B.10 Grades and Tests Used for Trajectory Growth in North Carolina and the 

Percent of Difference Expected to Be Closed Per Year ..................................... 125

Exhibit B.11 Illustration of Tennessee’s Method for Determining Whether a Student 

Is On-Track Toward Proficiency in Reading...................................................... 128

Exhibit C.1 Annual Measurable Objectives (AMOs) for Reading and Math, by 

State, 2006−07 and 2007−08 School Years........................................................ 129

Exhibit C.2 Number of Eligible Schools and Percentage in Which Math AMO Was 

Met Because of Growth Model Results, by All Students and 

Racial/Ethnic Reporting Groups, 2007–08......................................................... 130

Exhibit C.3 Number of Eligible Schools and Percentage in Which Math AMO Was 

Met Because of Growth Model Results, by Low-SES, SWD, and LEP 

Reporting Groups, 2007–08................................................................................ 131

Exhibit C.4 Numbers of Schools at Each AYP Status by NCLB School Improvement 

Status, by State, 2007−08.................................................................................... 132

Exhibit C.5 Numbers of Schools at Each AYP Status by School Poverty 

Concentration, by State, 2007−08....................................................................... 133

Exhibit C.6 Numbers of Schools at Each AYP Status by School Minority 

Concentration, by State, 2007−08....................................................................... 134
Evaluation of the Growth Model Pilot Project ix

Exhibit C.7 Numbers of Schools at Each AYP Status by School Urbanicity, by 

State, 2007−08 .................................................................................................... 135

Exhibit C.8 Percentage of Schools Meeting the AMO Using Growth-Only and 

Making AYP by Other Means After Growth-Only Is Applied, by State, 

2007–08............................................................................................................... 136

Exhibit D.1 Regression Coefficients and Framework Slopes for North Carolina Data ......... 138
Evaluation of the Growth Model Pilot Project x
Evaluation of the Growth Model Pilot Project xi

Acknowledgments

This report benefited from the efforts and thoughtful input of many individuals. Andrew Abrams 

at the U.S. Department of Education’s Policy and Program Studies Service (PPSS) directed the 

project and provided helpful guidance and encouragement at each stage. Daphne Kaplan, David

Goodwin, and Stephanie Stullich, also at PPSS, offered much constructive advice and feedback 

throughout the project.

The data analyzed in this report were drawn from the U.S. Department of Education’s EDFacts

archive and from each of the separate state departments of education participating in the Growth 

Model Pilot Project. We are grateful to Ross Santy and Susan Thompson-Hoffman for their 

generous help and patience with the EDFacts data. The Alaska, Arizona, Arkansas, Delaware, 

Florida, Iowa, North Carolina, Ohio, and Tennessee state departments of education were very 

responsive to our various requests for data and information about their pilot programs, and we 

would particularly like to acknowledge the help of Les Morse, Robert Franciosi, Denise Airola, 

Robin Taylor, Ed Croft, Thomas Deeter, Kenneth Barbour, Matt Cohen, and Dan Long.

We were very fortunate to receive thoughtful and timely reviews from several individuals both 

within and outside the U.S. Department of Education. In particular, we received extremely 

helpful guidance in the early stages of the project and detailed feedback on early drafts of the 

report from Patrick Rooney at the Office of Elementary and Secondary Education (OESE) within 

ED. We also thank Abigail Potts and Victoria Hammer of OESE for sharing their expertise in 

the design and review stages. In addition we would like to thank Alan Ginsburg, Jay Noell, and 

Emma Vadehra for their insight during initial drafting of the final report. Vadehra and Scott 

Sargrad provided very helpful comments on the later stages of the final report.

The project had a Technical Working Group (TWG) consisting of David Figilo, Thomas Fisher, 

Pete Goldschmidt, Rachel Quenemoen, and Keith Zvoch. The TWG provided guidance with the 

study design and analysis plans, and contributed many useful comments and corrections on the 

Interim Report. Pete Goldschmidt reviewed an early draft of this Final Report and provided 

many helpful suggestions. 

The Final Report differed from the Interim Report in incorporating data from the 2007–08 school 

year and conducting an in-depth analysis of differences among the pilot states, particularly the 

effects of using different types of growth models. The additional analysis benefitted greatly 

from the expertise of Andrew Ho of the Harvard Graduate School of Education and Katherine 

Furgol of the University of Iowa, who drafted much of Chapter 4 and provided excellent 

suggestions for improving much of the rest of the report. Jennifer Dunn at Measured Progress

also consulted on the design of the Final Report analyses. At NORC, the project was helped at 

several key junctures by the insightful perspectives of Stephen Raudenbush and Larry Hedges. 

Marie Halverson and Sarah Kay McDonald directed the project. Cindy Simko contributed 

greatly to collecting data from the states and organizing and implementing the many reviewer 

comments. The task of formatting this report in each of its iterations was handled with great skill 

and efficiency by Imelda Demus, Isabel Guzman-Barron, and Yajaira Gijon.
Evaluation of the Growth Model Pilot Project xii
Evaluation of the Growth Model Pilot Project xiii

Executive Summary

The U.S. Department of Education (ED) initiated the Growth Model Pilot Project (GMPP) in 

November 2005 with the goal of approving up to ten states to incorporate growth models in 

school adequate yearly progress (AYP) determinations under the Elementary and Secondary 

Education Act (ESEA). After extensive reviews, nine states were fully approved for the initial 

phase of the pilot project by the 2007–08 school year: Alaska, Arizona, Arkansas, Delaware, 

Florida, Iowa, North Carolina, Ohio, and Tennessee. Based on analyses of data provided by the 

U.S. Department of Education and by the pilot grantee states, this report describes the progress 

these states made in implementing the GMPP in the 2007–08 school year.

GMPP Objectives

Use of growth models for determining AYP is attractive to states and local districts because it 

offers a means to identify schools in which students are making progress even though they may 

not yet be reaching proficiency standards. Without recognition of the progress made by these 

students, these schools would be subject to school improvement actions that may not be 

appropriate in light of their demonstrable improvements.

The standard method of determining AYP has been the “status model,” in which school 

performance is mainly evaluated in terms of the proportion of students meeting or exceeding 

proficiency standards for reading and mathematics. The status model has been supplemented 

with “safe-harbor” provisions. Sometimes referred to as an improvement model, safe-harbor

recognizes schools that do not make AYP under the status model as making AYP if the 

percentage of non-proficient students decreased by 10 percent or more from the previous to the 

current school year.

In contrast, growth models measure how much students have gained from one year to the next 

using longitudinal records of individual student achievement in reading and mathematics. The 

models determine whether each student is “on-track” to reach or exceed the state’s grade-level 

proficiency cut points (or thresholds) on the annual tests of reading and mathematics within three 

or four years or by a specified grade level (usually grade eight or nine) as defined by the state’s 

particular growth model. For purposes of determining AYP, a student who is not proficient but 

on-track can be counted the same as a proficient student or as some fraction thereof.

Consistent with the general rules of ESEA accountability, the GMPP requires that the data on 

students’ proficiency and on-track to proficiency results are used to assess all students and each 

reporting subgroup: major racial or ethnic groups (American Indians, Asians, blacks, Hispanics, 

whites), students from low-income households, students with disabilities, and students with 

limited English proficiency. Each group must meet the same annual measurable objectives 

(AMOs) in order for a school to make AYP.
Evaluation of the Growth Model Pilot Project xiv

States Included in the Pilot Project

The GMPP began in 2005 with two states—North Carolina and Tennessee—approved to use 

growth models for ESEA accountability in the 2005–06 school year. The initial pilot project was 

limited to include no more than 10 states. The number approved to implement growth models 

under the pilot expanded to eight states in 2006–07 and nine states in 2007–08: Alaska, Arizona, 

Arkansas, Delaware, Florida, Iowa, North Carolina, Ohio, and Tennessee. This report focuses 

only on the nine states approved under the initial pilot project for 2007–08.

1

In December 2007, the U.S. Department of Education removed the initial pilot program’s cap of 

10 states and all states were able to use growth models for AYP determinations, pending 

approval by the Department. Following the change, six more states were approved under the 

open application process: Michigan and Missouri (beginning in the 2007–08 school year); and 

Colorado, Minnesota, Pennsylvania, and Texas (beginning in the 2008–09 school year). These 

six states were not part of the initial pilot project and were thus beyond the scope of this report.

Findings

Features of Growth Models Implemented by Pilot States

The growth models implemented under the GMPP were all designed to augment rather than 

replace the standard status model and safe-harbor provisions for determining school AYP. The 

growth models resulted in more schools making AYP than would have been the case using only 

status and safe-harbor.

Eight of the nine states used growth criteria only after schools failed to make AYP under the 

status and safe-harbor provisions. Delaware was the exception, applying growth results before 

status and safe-harbor. The designs of the pilots in the other eight states applied growth criteria 

only to ESEA reporting groups that did not reach their annual measurable objectives (AMOs) or 

obtain AYP via safe-harbor provisions. Furthermore, in all but Florida and Tennessee, the 

growth criteria were applied within the ESEA reporting groups only to the students who did not 

reach the proficiency threshold. The number of non-proficient but on-track students was added 

to the number of proficient students and the reporting group was counted as meeting the AMO if 

the total was high enough.

 

 1 An evaluation by the Office of Elementary and Secondary Education of growth model results in North Carolina 

and Tennessee for the 2005–06 school year is available at 

http://www.ed.gov/admins/lead/account/growthmodel/gmeval0109.doc.
Evaluation of the Growth Model Pilot Project xv

Comparing AYP by Growth with Status and Safe-Harbor

The growth models in the nine states that provided data resulted in some schools making AYP 

that would not have made AYP under status or safe-harbor alone (see Exhibit S.1).2

Ohio entered the program in 2007–08 and saw its growth model account for 50 percent of the 

schools that made AYP. Much of the high rate of making AYP by growth observed in Ohio is 

likely explained by their procedures for identifying non-proficient students as being on-track to 

proficiency. To reduce the risk of misclassifying those students as being not on-track, Ohio 

adopted a much more inclusive definition of on-track to proficiency and this had the 

consequence of helping relatively large numbers of schools make AYP that would not have done 

so without the growth model.

 Among all 

schools, 9 percent made AYP in 2007–08 uniquely because of the growth model (that is, they did 

not make AYP by status or safe-harbor). This compares with 3 percent of all schools making 

AYP due to growth during the 2006–07 school year. However, most of the schools that made 

AYP by growth were located in Ohio; excluding Ohio, only 2 percent of all schools in the other 

eight states made AYP by growth. The percentages of all schools that made AYP uniquely by 

growth varied widely among the states, ranging from literally 0 or 1 percent of all schools in 

Alaska, Arizona, and North Carolina to 34 percent of all schools in Ohio. The impact of GMPP 

varied little across the two school years with the exception of Iowa, which saw a decline from 11 

percent to just 2 percent of schools making AYP because of the pilot program.

 2 Delaware did not report AYP results in this form to the U.S. Department of Education because it elected to apply 

growth model results before status or safe-harbor. However, Delaware provided the authors with additional data 

identifying which schools uniquely made AYP because of the availability of the growth model and they are used in 

Exhibits S.1 and S.2.
Evaluation of the Growth Model Pilot Project xvi

Exhibit ES.1

Percentage of Schools That Made AYP Before and After the

Application of the Growth Model, in Nine States, in 2007–08 and 2006–07

44%

56%

59%

66%

75%

77%

56%

56%

67%

66%

19%

29%

66%

85%

31%

44%

33%

62%

83%

89%

9%

3%

0%

0%

<1%

0%

6%

8%

3%

3%

5%

5%

2%

11%

0%

1%

34%

n/a

2%

1%

0% 20% 40% 60% 80% 100%

All Schools, 2007–08

2006–07

Alaska, 2007–08

2006–07

Arizona, 2007–08

2006–07

Arkansas, 2007–08

2006–07

Delaware, 2007–08

2006–07

Florida, 2007–08

2006–07

Iowa, 2007–08

2006–07

North Carolina, 2007–08

2006–07

Ohio, 2007–08

2006–07

Tennessee, 2007–08

2006–07

Percent of Schools

Met AYP Before Growth Model Met AYP After Growth Model

Exhibit reads: Among all schools containing grade levels for which growth models were used for AYP 

determinations in the nine pilot states during the 2007–08 school year, 44 percent made AYP using 

status or safe-harbor methods. Another 9 percent made AYP because of the growth model.

As the bar chart illustrates, the pilot states also varied greatly in the proportion of their schools 

that made AYP under status or safe-harbor. The 1,246 schools making AYP uniquely by growth 

represented a percentage increase in the schools making AYP (column E in Exhibit S.2) of 20 

percent across all states, but this increase was largely comprised of 983 schools in Ohio and 153 

schools in Florida. Excluding Ohio, the overall percentage increase due to the GMPP in the 

other eight states was 5 percent.
Evaluation of the Growth Model Pilot Project xvii

Another measure of GMPP impact is the extent to which it reduced the number of schools that 

would not have made AYP had the growth model not been available. From this perspective, the 

number of schools that did not make AYP by either status or safe-harbor was reduced by 16 

percent overall because of the GMPP (column F in Exhibit S.2). The percentage reduction was 

by far the highest in Ohio (50 percent), followed by Arkansas (13 percent), and Tennessee (10

percent). Excluding Ohio, the overall percentage reduction due to the GMPP in the other eight 

states was 4 percent.

Exhibit ES.2

Number of Schools Making or Not Making AYP By Status, Safe-Harbor or Growth Model, 

Percentage Increase in Number of Schools That Made AYP Due to Growth, and 

Percentage Decrease in Number of Schools That Did Not Make AYP Due to Growth, by 

State, 2006–07

Pilot States

A:

Number of 

Schools 

Making AYP 

by Status or 

Safe-Harbor

B:

Number of 

Schools 

Making AYP

by Growth

C:

Number of 

Schools not 

Making AYP

D:

Total 

Number of 

Schools 

(A+B+C)

E:

Percentage 

Increase in 

Schools 

Making AYP 

Due to 

Growth

(B/A)

F:

Percentage 

Decrease in 

Non-AYP 

Schools Due 

to the Growth 

Model

(B/(B+C))

All Nine States 6,213 1,246 6,617 14,076 20% 16%

Alaska 292 0 203 495 0% 0%

Arizona 1,117 8 371 1,496 1% 2%

Arkansas 500 52 338 890 10% 13%

Delaware 123 5 55 183 4% 8%

Florida 632 153 2,495 3,280 24% 6%

Iowa 721 23 354 1,098 3% 6%

North Carolina 737 0 1,612 2,349 0% 0%

Ohio 961 983 984 2,928 102% 50%

Tennessee 1,130 22 205 1,357 2% 10%

Exhibit reads: The 1,246 schools that made AYP by growth increased the number of schools making AYP from 

6,213 to 7,459 schools, which was a percentage increase of 20 percent. Of the schools that did not make AYP 

under either status or safe-harbor (1,246+6,617=7,863), the growth model decreased the non-AYP total by 16 

percent.

Source: U.S. Department of Education, EDFacts and the Delaware state department of education.

Impact of Growth Models on AYP Rate Among High-Poverty Schools

Schools serving disadvantaged populations have been found to make AYP at much lower rates 

than those serving more affluent populations (U.S. Department of Education, 2007). The growth 

model pilots may reduce these associations to some extent by identifying high-growth schools 

serving low-income and minority communities.
Evaluation of the Growth Model Pilot Project xviii

The results of this analysis showed that schools serving economically disadvantaged student 

populations in all pilot states except for Arkansas were more likely than more-advantaged 

schools to make AYP by growth. Across all nine states, the percentage increase in the number of 

high-poverty schools making AYP as a result of the growth model being available was 20

percent (994 schools instead of 826 schools), compared to 18 percent among low-poverty 

schools (2,298 schools instead of 1,946 schools). However, the percentage increases among 

high-poverty schools were much greater than those among low-poverty schools in Florida (77 

instead of 45 for high-poverty schools compared to 300 instead of 272 for low-poverty schools),

Ohio (172 instead of 64 for high-poverty schools compared to 771 instead of 458 for lowpoverty schools), and Tennessee (280 instead of 264 for high-poverty schools compared to 120 

instead of 119 for low-poverty schools). 

Hypothetical Results of Using Growth Models Instead of Status and Safe-Harbor

The GMPP application guidelines noted that states could use growth as the primary 

accountability indicator for all schools, including those that made AYP by status. This growthonly model would lower AYP percentages by excluding proficient students who are not on-track 

to maintain proficiency due to declines. A possible advantage of using a growth-only model is 

that it could provide better predictions of whether students will reach or maintain proficiency 

goals within the pilot’s time frame (three or four years, or by grade 8 or 9 in most of the states). 

The student data were used to assess the extent to which the schools that made AYP by status 

and safe-harbor (examined separately) would also have met or exceeded their AMO for reading 

and mathematics proficiency if the growth criteria of on-track-to-proficiency were used instead 

of the status or safe-harbor criteria. 

Overall, 62 percent of the schools that made AYP by status criteria in the nine states also would 

have met their reading and mathematics AMOs strictly by using the growth criteria. Results 

varied widely among the states, ranging from only 46 percent in Arizona and 47 percent in 

Arkansas and North Carolina to 75 percent or more in Ohio, Delaware, and Tennessee. These 

findings provide some evidence that despite the low rates of making AYP by growth found in 

most states, many schools had sufficiently high rates of students being on-track to reach or 

maintain proficiency to meet their AMOs using growth only. Under the normal practice of 

applying status before growth, the large numbers of schools that could have met their AMOs by 

growth-only were obscured. The percentage of schools that made AYP by safe-harbor and that 

also met or exceeded their reading and mathematics AMOs under the growth-only criteria was

much lower (28 percent overall). Across the eight states with safe-harbor schools (Delaware had 

none), the percentages did not exceed 30 percent in any state except Arkansas (64 percent) and 

Ohio (45 percent). This relatively low level of overlap of safe-harbor and growth-only outcomes 

indicates that, despite an ostensibly similar purpose of identifying progress toward proficiency 

goals among schools that have not reached status model AMOs, the two methods often led to 

different results, possibly because safe-harbor measures school progress while growth-only 

measures individual student growth. 
Evaluation of the Growth Model Pilot Project xix

Effects of Growth Model Types on Student and School Outcomes

The growth models implemented by the nine pilot states generally grouped into three types: 

transition matrix models (which evaluate student progress from year to year in terms of a 

relatively small set of discrete performance levels), trajectory models (which use the gap 

between a baseline test score and a performance standard several years out to calculate the 

amount of growth required to become proficient), and projection models (which use current and 

past test scores to statistically predict performance several years ahead). Effects of model types 

were assessed by using data from a single state and applying generic versions of the models to 

those data. 

Results show that the projection model functions in stark contrast with transition matrix and 

trajectory models in terms of identifying students as on-track to reach or maintain proficiency. 

The transition matrix model acts as a coarse, categorical approximation of the trajectory model, 

with agreement rates on classifications of students as on-track or not on-track to proficiency of 

over 90 percent. The overlap is greatest when time horizons for the trajectory model are long or 

when the number of categories for the transition matrix model is large. In contrast, when a 

projection model classifies a non-proficient student as on-track, the probability that a transition

matrix or trajectory model agrees is near zero, and vice versa. However, agreement rates for 

non-proficient students are around 60 percent due to the three types of models agreeing about 

non-proficient students who are not on-track.

The underlying reasons for the contrast between the projection model and the transition matrix or

trajectory models relate to differing objectives guiding the models. Projection models are 

designed to maximize the accuracy of a prediction about whether a student will meet or exceed 

future proficiency standards. From a statistical standpoint, the best predictor of future 

achievement is past achievement. As a result, relatively few students with records of low 

achievement but evidence of improvement are predicted to meet or exceed future proficiency 

standards, while students with records of high achievement but evidence of slipping are very 

likely to be predicted to meet or exceed future proficiency standards. Transition matrix and 

trajectory models, in contrast, are designed to identify specific growth targets that each student 

must attain in order to meet or exceed the future proficiency standards, given his or her 

benchmark score (usually from the last test administration). Reflecting these different guiding 

objectives, projection models classify fewer previously non-proficient students who are gaining 

as on-track than the other models but a larger number of previously proficient decliners than the 

other models. For status-plus-growth models (which use growth model results only for nonproficient students), projection models will have the least impact, affecting only 10 to 20 percent

of eligible (non-proficient) students while transition matrix and trajectory models affect over 20

percent. States with higher proficiency cut scores will have heightened differences between the 

model types and a lower proportional impact of projection models on eligible students. 

In simulations of school AYP determinations, the models do not yield large differences in the 

percentages of schools making AYP when non-proficient students who are on-track to 

proficiency according to each type of model are added to the numbers of students meeting or 

exceeding the proficiency cut point (i.e., status-plus-growth). The models differ much more 
Evaluation of the Growth Model Pilot Project xx

when a growth-only calculation is made first, followed by status-plus-growth calculations. 

Under the growth-only simulation using a realistic proficiency cut score and standard AMOs, 

very few schools would make AYP with a trajectory or transition matrix model while most 

schools would make AYP with a projection model.

Predictive Accuracy of the Different Types of Growth Models

The growth models implemented in the GMPP have two important goals that turn out to be 

somewhat contradictory: to predict as accurately as possible whether non-proficient students will 

attain proficiency within a delimited time frame, and to identify as clearly as possible the 

performance levels students must achieve at each grade level in order to attain proficiency within 

the designated time frame. A comparison of the student on-track versus not-on-track 

determinations from the three generic types of growth models using the same data shows that the 

projection model has the highest correct classification rates for future proficiency: over 80

percent. These rates are 5 to 20 percentage points higher than trajectory and transition matrix

models, depending on the grade level and proximity to the growth model time limit.

While the greater predictive accuracy of the projection model may be cited as an advantage, the 

simpler trajectory and transition matrix models may provide clearer guidance to schools, 

teachers, students, and their parents about the amount of growth needed to reach or maintain 

proficiency from year-to-year. This is because the simpler models identify a level of 

achievement that each student must attain at each grade in order to be on-track to reach or 

maintain proficiency, while the projection model cannot easily identify intermediate achievement 

targets for individuals because of the complex statistical apparatus used to predict future scores. 

Effects of Alternative Standards of Adequate Yearly Growth

The GMPP core principles dictated that the pilot growth models must be designed to assess 

progress toward grade-level proficiency. However, attaining the on-track designation can 

require very large annual growth increments for students who start at low levels of achievement. 

At the other extreme, for students who start at high levels, attaining the on-track designation may 

be possible with no growth at all. In light of these shortcomings of proficiency-based growth 

standards and current policy interest in alternatives, an analysis of the effects of other ways of 

assessing growth was undertaken.

An alternative criterion-referenced standard of annual growth that can be used with vertical test 

score scales is the difference between the proficiency cut scores in successive grade levels. 

Students gaining that amount or more would be considered to make “adequate yearly growth” 

regardless of whether they are proficient or on-track to become proficient. A simulation shows 

that the overall percentages of students meeting that alternative standard of growth are lower 

than the percentage of proficient students in both reading and mathematics, but that the 

percentages of non-proficient students meeting the alternative standard are higher than that 

meeting the GMPP standard. Consequently, adding the non-proficient students who met the 

alternative growth standard to the pool of proficient students would increase the overall rates of 

students who are arguably performing adequately (i.e., proficient or making reasonable progress 
Evaluation of the Growth Model Pilot Project xxi

over the past year). As expected, these methods are dependent on the location of proficiency cut 

scores up the vertical scale.

Qualifications and Implications 

The results of this analysis show that use of growth models generally added to the number of 

schools making AYP but that the numbers were not large in almost all pilot states. 

The GMPP growth models were required to measure students’ growth toward meeting or 

exceeding proficiency standards in reading and mathematics. This means that substantial 

individual student performance improvements that do not reach the growth models’ proficiency 

targets (generally within three or four years, or by grades 8 or 9) or subgroup- and schoolaggregate student performance improvements that do not reach the proficiency targets (AMOs)

are not recognized by the GMPP growth models. However, if ESEA regulations were revised so 

that a broader range of student gains were recognized as acceptable, more students and schools 

would be identified as making adequate progress than is currently the case. Such other targets 

generally involve use of a more finely graduated set of performance outcomes than proficiency 

or on-track to proficiency.

The generally low rates of making AYP by growth also reflect the impact of the various other 

(nongrowth) methods for determining AYP available in those states for schools to make AYP by 

status and safe-harbor (e.g., confidence intervals and multiyear averaging), such that the status 

and safe-harbor methods picked up schools which would have made AYP by growth had those 

various provisions not been available. Those additional nongrowth ways of making AYP are 

generally intended to reduce the chance of misclassifying schools as not making AYP but may 

obscure the extent (or lack) of student progress in the schools.

Within the current regulatory context of the ESEA, an implication of the results presented here is 

that states could clarify each school’s progress by applying growth criteria to all their schools 

and subgroups before status and safe-harbor. The main advantage of applying the growth model 

before the status and safe-harbor models is that it would identify schools that are realizing 

adequate progress toward universal proficiency. This would clearly distinguish those schools 

from schools making AYP under status or safe-harbor criteria but not realizing growth sufficient 

to continue meeting their AMOs. Identifying such schools would serve as an early warning 

mechanism of possible problems. The exploratory analyses in this report also indicate that 

applying growth criteria before safe-harbor could usefully reclassify many (28 percent overall) of 

the current safe-harbor schools as making AYP by growth, and would clearly identify those that 

are not on-track to proficiency and thus likely headed for improvement status in the near future. 

Another reporting option is to classify each school in terms of both growth and status. Schools 

making AYP would be distinguished as making AYP by both growth and status, by growth only, 

by status only, by a mix of status and growth, or by safe-harbor only. This would have the 

advantage of uniquely identifying different sets of schools (those making AYP in terms of both 

growth and status).
Evaluation of the Growth Model Pilot Project xxii

This study has also shown that the types of growth models states select for federal accountability 

purposes are consequential and raise some potentially difficult theoretical questions for 

policymakers. Projection models are likely to be more accurate than transition matrix or 

trajectory models in terms of predicting students’ future attainment of proficiency targets, but the 

simpler trajectory and transition matrix models may provide clearer guidance on annual 

achievement goals to schools, teachers, students, and their parents. 

The projection model is explicitly designed to provide probabilistic predictions whereas the other

models do not. As a probabilistic estimator, the projection model carries a measure of 

uncertainty for each student’s predicted score. An important issue illustrated by the Ohio model 

is whether to adjust a student’s predicted score for the uncertainty and, if so, to what extent. The 

adjustment used by Ohio (adding two standard error units to each student’s predicted score) was 

selected in order to make it highly unlikely that a student who actually was on-track was 

misclassified as not on-track. However, the adjustment also had the effect of classifying a much

higher percentage of non-proficient students as on-track than the other pilot states. Careful 

consideration of the trade-off between false-negatives and false-positives is needed when 

adjustments for uncertainty are made to statistically derived growth model results. 
Evaluation of the Growth Model Pilot Project 1

I. Introduction

A key goal of the 1994 reauthorization of the Elementary and Secondary Education Act (ESEA)

was to introduce a standards-based accountability system that “required states to define criteria 

for measuring adequate yearly progress (AYP) in school performance for Title I schools and 

districts.”3 States were given considerable latitude in how to determine AYP, with the majority 

relying on various types of aggregate school-improvement models rather than setting absolute 

proficiency targets for students.

4

With the increasing availability of statewide longitudinally linked student performance records 

since 2001, it became possible for some states to measure student growth and use those data for 

accountability purposes. The U.S. Department of Education initiated the Growth Model Pilot 

Project (GMPP) in November 2005 with the goal of approving up to ten states to incorporate 

growth models in school AYP determinations under ESEA. Growth models are defined as 

complements or alternatives to the standard status model for determining school AYP. The 

status model bases AYP on the proportion of a school’s students attaining proficiency in reading 

and mathematics in a given year. Growth models, in contrast, base AYP in part on the 

proportion of individual students who are making sufficient annual progress to reach grade-level 

proficiency within a specific time horizon of three to five years or by grades 7, 8, or 9. Growth 

models promise to provide a fuller understanding of school effectiveness and the progress each 

school’s students are making toward their proficiency goals. The main objectives of the GMPP 

are to help states develop and implement models for determining school-level AYP that 

incorporate measures of student growth. 

 However, the 2001 reauthorization of ESEA established the 

primacy of absolute proficiency standards by requiring states to: (a) develop grade-level specific 

proficiency standards in both reading and mathematics for grades 3–8 and one or more high 

school grades, and (b) to assess the performance of all students in those grades each year. The 

states’ targeted percentages of students scoring at or above the proficiency standards increase at 

least every three years with all students expected to be proficient by 2014. Based on its students’

scores, every public school is evaluated to determine whether or not it is making AYP, and 

consequences are applied to schools not making AYP for more than two consecutive years. 

Scope of This Report

This final evaluation of the GMPP is restricted to the nine states approved for participation in the 

initial pilot project during the 2007–08 school year. The states approved for participation in the 

initial pilot project were: North Carolina and Tennessee (approved for implementation 

 3 U.S. Department of Education, Office of the Under Secretary, Policy and Program Studies Service (2004), 

“Evaluation of Title I Accountability Systems and School Improvement Efforts (TASSIE): First-Year Findings,”

Washington, D.C. (retrieved December 2009 from http://www.ed.gov/rschstat/eval/disadv/tassie1/index.html). 

4 Ibid. Examples of school-improvement models under the 1994 reauthorization included making AYP if any growth 

in school-average achievement, or the percentage of proficient students, occurred from year-to-year or if the gap 

between low- and high-achieving students was reduced by a given percentage each year. However, a few states 

adopted absolute standards of proficiency and measured school progress in ways very similar to those mandated by 

the 2001 reauthorization of ESEA.
Evaluation of the Growth Model Pilot Project 2

beginning in the 2005–06 school year); Alaska, Arizona, Arkansas, Delaware, Florida, and Iowa 

(approved for implementation beginning in the 2006–07 school year); and Ohio (approved to 

begin implementation in the 2007–08 school year). While the GMPP began in 2005 with a goal 

of approving up to 10 states, the Department made the option of using growth models available 

to all states in December 2007, substantially expanding the scope of the pilot. In June 2008 and 

January 2009 Secretary Spellings announced the approval of six additional growth models, in 

Michigan and Missouri (approved for use in the 2007–08 school year), and Colorado, Minnesota, 

Pennsylvania, and Texas (beginning in the 2008–09 school year), but those states were not part 

of the initial pilot project and thus are beyond the scope of this report.5

The report is designed to answer three questions:

How have states in the pilot project implemented growth models?

How does each pilot state’s growth model affect the number and kinds of schools that make 

AYP?

What are the implications of the pilot project experience for extending and strengthening 

growth models within the context of ESEA? 

The remainder of this chapter describes how the GMPP models compare with status models in 

approaches to evaluating student achievement. Chapter II considers, for each of the nine pilot 

grantee states in the 2007–08 school year, the impact of the state’s GMPP model on its AYP 

determinations. Chapter III examines the effect of the growth models on AYP outcomes in 

different types of schools.

The implications of the initial pilot project experiences for future efforts to use growth models 

are developed in Chapter IV. Analyses in that chapter address a number of hypothetical 

questions about how results might change if the data collected as part of the pilot project were 

used differently. This section also considers how the type of growth model selected affected ontrack determinations and how the required focus on grade-level proficiency affects AYP 

outcomes. This section also evaluates the adequacy of state longitudinal data systems for 

implementing growth models.

Some of the analyses in Chapter IV address hypothetical questions about alternatives to current 

ESEA regulations for the Growth Model Pilot Project and would thus only become practical if 

the regulations were changed. While these analyses are part of the contractual scope of work for 

the evaluation project reported on here, they do not reflect any type of an endorsement by ED of 

the alternatives analyzed.

 5 See “Secretary Spellings Invites Eligible States to Submit Innovative Models for Expanded Growth Model Pilot”

(retrieved September 2010 from http://www2.ed.gov/news/pressreleases/2007/12/12072007.html ) and “U.S. 

Secretary of Education Margaret Spellings Approves Additional Growth Model Pilots for 2007–08 School Year” 

(retrieved June 2008 from http://www.ed.gov/news/pressreleases/2008/06/06102008.html) and “Secretary Spellings 

Approves Additional Growth Model Pilots for 2008–09 School Year” retrieved January 2009 from 

http://www.ed.gov/news/pressreleases/2009/01/01082009a.html).
Evaluation of the Growth Model Pilot Project 3

The Status Model of Accountability Under ESEA

Under ESEA as amended in 2001, each state revised its standards-based system of student 

achievement measures and targets and began conducting annual assessments of proficiency 

levels to determine whether its schools and districts were making AYP.6

Each step in the path to achieving universal proficiency in reading and mathematics under ESEA

is known as the “annual measurable objective” or “AMO.”

 Each school has a 

certain percentage of students who score proficient or higher each year on the mathematics and 

reading or language arts achievement tests, and this constitutes an annual measure of a school’s 

performance. That percentage is expected to reach 100 percent by the end of the 2013–14 school 

year in incremental steps. In addition, ESEA requires each school to meet or exceed statewide 

standards on one or more “other academic indicators,” typically defined in terms of average daily 

attendance for elementary schools and graduation rate for high schools.

7

In order for the school to make AYP under the ESEA status model, several conditions must be 

met. These conditions are required by the law and are intended to improve the reliability and 

validity of the accountability results. First, the school must test at least 95 percent of its students

in each ESEA reporting group in both reading or language arts and mathematics. These ESEA

reporting groups consist of all students plus major racial and ethnic subgroups, students with 

disabilities, limited English proficient students, and students from low-income households. 

Within each school, a reporting group may be excluded from federal accountability requirements 

if the number of “full academic year” students from that group is below a minimum “n” size. 

Most states define a full academic year as starting in the fall when enrollments are finalized 

(typically around Oct. 1) and extending through the end of the testing window in the spring, 

while the state-defined minimum n sizes for reporting groups range from a low of no minimum 

to 100 students. Second, the percentage of tested students scoring proficient or higher must meet 

or exceed the AMO in both subjects for eligible reporting groups. These percentages are 

calculated only for students enrolled for the full academic year. If a single subgroup fails to 

achieve the AMO, the school does not make AYP. 

 The AMO is the standard that 

schools and districts use to determine whether or not they are making AYP. AMO trajectories 

are not uniform across states, including the states in this study. Some increase in a consistent 

linear fashion toward 2014, while others increase more in the years closer to 2014 than in those 

closer to the 2002 start of the AYP requirements. 

In order to reduce the chances of incorrectly classifying schools as not making AYP, the states 

are allowed to apply additional steps within the status model. If any one or more of the ESEA

reporting groups did not make AYP, the school may:

 6 See Elementary and Secondary Education Act, Title I – Improving the Academic Achievement of the 

Disadvantaged, Section 1111, paragraph (b)(2) retrieved October 2010 from 

http://www2.ed.gov/policy/elsec/leg/esea02/pg2.html#sec1111 . 7 AMO trajectories are defined by the states. They have levels for each school year that increase at least every three 

years and can be different within each year across grades and subjects. States were required to follow strict statutory 

requirements in setting their initial AMOs. 
Evaluation of the Growth Model Pilot Project 4

• Apply a confidence interval to the group’s percent proficient and compare the upper 

bound to the AMO. Analogous to the margin of error typically reported with results from 

political and other opinion polls, a confidence interval represents the range of values 

within which the true value is expected to fall for a given level of statistical certainty

(e.g., 95 percent). The higher the standard of certainty asked for, the wider the 

confidence interval. For a given standard of certainty, the confidence interval narrows as 

the student count grows. If the higher limit of that confidence interval is greater than the 

AMO, then the subgroup is considered to make AYP.

• Average the test results for the group over two or three years and compare the average to 

the AMO (this is often referred to as “multiyear averaging”).

• Apply safe-harbor, whereby the group makes AYP if the percentage of non-proficient 

students in the group decreased by 10 percent or more from the prior year (e.g., a decline 

from 30 percent to 27 percent non-proficient). 

• Apply safe-harbor but assess whether the reduction in the percentage non-proficient was 

10 percent or more from the average percentage non-proficient over the prior two or three 

years.

Variations of this basic method for determining school AYP are used by all states and are 

collectively referred to as the status model. A simplified version of the decision tree (excluding 

the full academic year, minimum n, confidence intervals, and multiyear averaging conditions) is 

illustrated in Exhibit 1.

Exhibit 1

Determining AYP Under the Status Model

Exhibit reads: A school’s AYP under the status model is determined by first assessing whether 

95 percent of the students in each reporting group took the reading and the mathematics tests. If 

so, the percentages of the test-takers that scored at or above the proficiency cut scores are 

calculated and are compared to the AMO. If the percentage proficient for any reporting group is 

below the AMO, then the group’s percentage non-proficient is compared to its percentage nonproficient from the prior school year. If the current percentage non-proficient is 10 percent or 

more lower than the prior year, the group makes AYP by safe-harbor. 
Evaluation of the Growth Model Pilot Project 5

Limitations of the Status Model

One characteristic of the status model is that it does not recognize real improvements in student 

achievement unless they result in higher percentages of students meeting or exceeding 

proficiency standards in a given year. Schools with many students improving but still falling 

short of the proficiency standard may not meet their AMO and thus will not make AYP. 

Conversely, schools with many students at or above the proficiency standard will still make AYP

even if few of these students improve from year-to-year. 

The fact that schools are evaluated strictly in terms of the numbers of proficient students, with no 

credit given for improvement short of or over and above proficiency, has raised concerns that, in 

the short term, instructional resources may be focused on students who are closest to the 

proficiency threshold (“bubble students”). Students less likely to attain proficiency from a given 

amount of instructional effort—e.g., those farthest below the proficiency threshold—may receive 

less attention. Studies investigating this hypothesis have come to mixed conclusions, however, 

and rising AMOs make this strategy less relevant because all students must be proficient by 2014 

in order for a school to make AYP. On the other side of the ledger, high-scoring students may

receive less attention because there are no statutory consequences for failing to improve 

achievement among those already proficient.

8

Another characteristic of the status model is that it does not take account of changes in a school’s 

student composition from one year to the next. Thus a school classified as not making AYP in 

one year could be judged to make AYP in the next if more proficient students enrolled or if lessproficient students left. Conversely, a school making AYP one year may not reach the AMO 

standard the following year if its student composition shifted the other way. 

This possible tendency is tied to the use of a 

minimum proficiency threshold and would thus not be affected by the 2014 target. 

The Growth Model Pilot Project (GMPP)

The U.S. Department of Education (ED) initiated the Growth Model Pilot Project (GMPP) in 

November 2005 with the goal of approving up to ten states to incorporate growth models in 

school AYP determinations under ESEA. Growth models are defined as complements or 

alternatives to the standard status model for determining school AYP; they base AYP on some 

measure of how much students have gained from one year to the next. 

Growth models are intended to recognize schools’ progress moving students toward proficiency. 

As suggested above, student growth patterns in a school may overlap with or diverge from 

 8 Research on the extent to which schools have adopted these sorts of strategies has been conducted by Naomi 

Chudowsky, Victor Chudowsky, and Nancy Kober (2009), “Is the Emphasis on ‘Proficiency’ Shortchanging 

Higher- and Lower-Achieving Students?” Retrieved July 2009 from http://www.cepdc.org/index.cfm?fuseaction=document_ext.showDocumentByID&nodeID=1&DocumentID=280; Derek Neal and 

Diane Whitmore Schantzenbach (August 2007), NBER Working Paper 13239 “Left Behind By Design: Proficiency 

Counts and Test-Based Accountability” retrieved September 2008 from http://www.nber.org/papers/w13293.pdf ;

and Jennifer Booher-Jennings (2005), “Below the Bubble: ‘Educational Triage’ and the Texas Accountability 

System,” American Educational Research Journal, 42: 231–268. 
Evaluation of the Growth Model Pilot Project 6

assessments of students’ proficiency based on the status model; these are illustrated in Exhibit 2. 

Ideally, all schools would be in cell A, meeting the status requirements and realizing high rates 

of annual growth. The basic goal of the GMPP is to identify schools in cell B, that is, schools 

with high numbers of students making progress but not yet attaining the grade-level proficiency 

thresholds necessary to meet AYP standards. Schools in cell C—those with low rates of student 

growth but still making AYP—are also of interest to the GMPP but, as will be discussed further 

at various points in this report, were not targeted in the project. The overarching goal of ESEA as 

implemented by the GMPP is to ensure that no schools are in cell D, not meeting the status 

model requirements and not making sufficient gains to be on-track to meet the requirements in 

the near future. 

Exhibit 2

Relationship of Growth and Status Models for Assessing Achievement Proficiency

School-level Growth in Achievement

School AYP Designation Under the Status Model

Made AYP Did Not Make AYP

High rates of growth Cell A Cell B

Low rates of growth Cell C Cell D

The U.S. Department of Education used a rigorous peer review process to evaluate the adequacy 

of the technical aspects of the proposed models and to ensure that the models aligned with seven 

core principles.9

The second principle stipulates that growth expectations cannot be based on student background 

or school characteristics. This principle is consistent with the ESEA rule that proficiency targets 

must be the same for all students in a given grade and cannot be modified for different kinds of 

students or schools. In the growth model context, this excludes use of the “relative-growth”

models permitted under the 1994 amendments that gave AYP credit to schools or subgroups 

strictly on the basis of realizing average or even higher-than-average annual growth rates. The 

key criterion under the 2001 reauthorization of ESEA is “meeting grade-level proficiency.”

Students scoring below proficiency must not only gain more per year than one grade-level 

equivalency, but those gains must also point to attaining proficiency standards within a specified 

 These core principles required all pilot states to set annual “growth targets” for 

ensuring universal grade-level proficiency by 2014 and to track individual students across 

schools and measure their progress across grades in both reading and mathematics. The first 

principle requires that the growth model, like the status model, be applied to each targeted 

subgroup as well as all students in the school. This means that growth outcomes are to be 

monitored separately, or “disaggregated,” for major racial and ethnic groups, limited English 

proficient (LEP) students, special education students, and low-income students. 

 9 See U.S. Department of Education (Nov. 18, 2005) “Press Release: Secretary Spellings Announces Growth Model 

Pilot, Addresses Chief State School Officers’ Annual Policy Forum in Richmond” (retrieved May 2008 from 

http://www.ed.gov/news/pressreleases/2005/11/11182005.html), and U.S. Department of Education (July 2007) “No 

Child Left Behind fact sheet: Growth Models—Ensuring Grade-Level Proficiency for All Students by 2014”

(retrieved May 2008 from http://www.ed.gov/admins/lead/account/growthmodel/proficiency.pdf ).
Evaluation of the Growth Model Pilot Project 7

time frame.10

Exhibit 3

Seven Core Principles of the Growth Model Pilot Project

 A full list of the seven core principles of the Growth Model Pilot Project is 

provided in Exhibit 3.

States approved for participation in the GMPP were required to meet seven core principles in the ESEA

accountability plans they submitted for incorporating growth models in their AYP measurements:

1. Ensure that all students are proficient by 2014, and set annual goals to ensure that the achievement gap is 

closing for all groups of students;

2. Set expectations for annual achievement based on meeting grade-level proficiency, not on student 

background or school characteristics; 

3. Hold schools accountable for student achievement in reading or language arts and mathematics;

4. Ensure that all students in tested grades are included in the assessment and accountability system, hold 

schools and districts accountable for the performance of each student subgroup, and include all schools and 

districts; 

5. Include assessments in each of grades 3–8 and in high school for both reading or language arts and 

mathematics, and ensure that they have been operational for more than one year and receive approval 

through the NCLB peer review process. The assessment system must also produce comparable results from 

grade to grade and year to year; 

6. Track student progress as part of the state data system; and 

7. Include student participation rates and student achievement on a separate academic indicator in the state 

accountability system.

Source: See “Peer Review Guidance for the NCLB Growth Model Pilot Applications,” January 25, 2006 (retrieved 

May 2008 from http://www.ed.gov/policy/elsec/guid/growthmodelguidance.pdf).

Other significant features of pilot growth models were allowed to vary, as long as the technical 

specifications passed review by a panel of nationally recognized experts.11

 10 This guiding principle also precludes the use of growth models referred to as “value added” models (VAM), at 

least when they include statistical adjustments for student social background variables or school characteristics. In 

general terms, value-added formulations seek to separate the effects of various factors on student growth in order to 

make attributions about the marginal effectiveness of the factors on achievement gains. Inputs are variously defined, 

depending on the purpose of the analysis, but often include student social background variables, teachers, and 

instructional programs. If used for evaluation of schools, value-added models estimate growth for a common 

“average type” of student in each school and the schools are then compared in terms of that backgroundstandardized estimate. In that sense, VAM models are typically measures of relative school effectiveness, not 

student proficiency or growth toward proficiency. The latter is the purpose of GMPP, and VAM are generally 

inappropriate for that purpose.

 Reviewers had a 

series of meetings to discuss the Peer Review Guidance document and the Department’s 

expectations for the process. Proposals that passed an initial review by Department staff and a 

round of clarification by states were forwarded to the peer reviewers. The Department then set 

up conference calls between the states and the peer review team, after which the reviewers met 

again to discuss the proposals and make recommendations for use by the secretary in deciding 

which proposals to approve for the pilot project. 

11 See “Peer Review Guidance for the NCLB Growth Model Pilot Applications,” January 25, 2006 (retrieved May 

2008 from http://www.ed.gov/policy/elsec/guid/growthmodelguidance.pdf).
Evaluation of the Growth Model Pilot Project 8

Of the 20 states that submitted proposals by February 2006, 13 asked for approval to use growth 

models for immediate use in the 2005–06 school year while the others proposed to start in 

2006–07. Eight of these proposals passed the initial evaluation, and revised proposals were 

forwarded to the peer reviewers.12 Two states, North Carolina and Tennessee, received approval 

to use growth models for 2005–06, while Delaware, Arkansas, and Florida revised their 

proposals and were approved in a subsequent peer review to implement their proposed models 

for the following school year (2006–07). A second round of proposals was solicited in October 

2006. Six previous applicants submitted revised proposals along with submissions from three 

new states.13 Of these, Alaska, Arizona, and Iowa received immediate or conditional approval to 

use their growth models beginning with the 2006–07 school year, while Ohio was approved to 

begin in the 2007–08 school year.14

The approved pilot states were all required to calculate whether each student in each target grade 

level was on-track to be proficient within a specified number of years or by a particular grade 

level. The growth models determined whether each student was on-track to reach or exceed the 

state’s grade-level proficiency cut scores on the annual tests of reading and mathematics within 

three or four years or by a specified grade level (usually grade eight or nine) as defined by the 

state’s particular growth model. Students who were on-track to be proficient could be counted as 

proficient for the purposes of determining AYP. However, the states were given a variety of 

options on how to incorporate the student growth indicator in their AYP determinations.15

• only the growth measure to calculate the percentage on-track to proficiency for AMO 

assessment; 

These 

included using:

• both the status and the growth measures to calculate the percentage proficient or on-track

for AMO assessment;

 12 These reviewers included Eric Hanushek, Stanford University, Chris Schatschneider, Florida State University, 

David Francis, University of Houston, Margaret Goertz, University of Pennsylvania, Kati Haycock, The Education 

Trust, William Taylor, Citizens Commission on Civil Rights, Sharon Lewis, Council of Great City Schools (retired), 

Robert Mendro, Dallas Independent School District, Jeff Nellhaus, Massachusetts Department of Education, and 

Mitchell Chester, then at the Ohio Department of Education.

13 See “Secretary Spellings Approves Additional Growth Model Pilots for 2006–2007” (retrieved June 2008 from 

http://www.ed.gov/news/pressreleases/2006/11/11092006a.html). This panel of reviewers included Anthony Bryk, 

Stanford University, Harold Doran, American Institutes for Research, Chrys Dougherty, National Center for 

Educational Accountability, Lou Fabrizio, North Carolina Department of Public Instruction, Tom Fisher, 

Independent Consultant, Pete Goldschmidt, University of California at Los Angeles, Sharon Lewis, Council of Great 

City Schools (retired), Margaret McLaughlin, University of Maryland, Robert Mendro, Dallas Independent School 

District, Jeff Nellhaus, Massachusetts Department of Education, Ann O’Connell, Ohio State University, Dianne 

Piché, Citizens Commission on Civil Rights, Sandy Sanford, Riverside County Office of Education, Chris 

Schatschneider, Florida State University, William Taylor, Citizens Commission on Civil Rights, and Martha 

Thurlow, University of Minnesota.

14 In December 2007, the Department moved to expand participation in the growth model project and extended an 

invitation to all states to apply for inclusion. At the time of writing, 15 states have been approved to implement 

growth models in AYP determinations.

15 See “Peer Review Guidance for the NCLB Growth Model Pilot Applications” (retrieved May 2008 from 

http://www.ed.gov/policy/elsec/guid/growthmodelguidance.pdf).
Evaluation of the Growth Model Pilot Project 9

• the status measure to calculate the percentage proficient, applying safe-harbor provisions 

if needed, and using the growth measure either in conjunction with the status measure or 

alone if AYP was not met with safe-harbor; and

• safe-harbor provisions and the growth measure for AMO assessment.

While the approved models differ from one another in a number of important ways, all use statespecific assessment data to measure student progress and proficiency, and the method of 

incorporating growth outcomes in AYP determinations was generally the same. Eight of the 

nine pilot states proposed to apply growth criteria only after schools failed to make AYP under 

the status and safe-harbor provisions rather than determining AYP solely on the basis of student 

growth. More specifically, the designs of the pilots in these eight states applied growth criteria 

only to those students who were members of ESEA reporting groups that did not reach their 

AMOs or attain AYP via safe-harbor provisions. This basic method of augmenting status with 

growth results is shown in Exhibit 4 (again, this is a simplification in that provisions for using 

confidence intervals and multiyear averaging prior to applying growth results are not 

represented).

Exhibit 4

Determining AYP Under Status, Safe-Harbor, and Growth

Exhibit reads: School AYP under the status-plus-growth model is determined by first assessing 

whether 95 percent of the students in each reporting group took the reading and the mathematics tests. 

If so, the percentages of the test-takers that scored at or above the proficiency cut scores are calculated 

and are compared to the AMO. If the percentage proficient for any reporting group is below the AMO, 

then the group’s percentage non-proficient is compared to its percentage non-proficient from the prior 

school year. If the current percentage is 10 percent or more lower than the prior year, the group makes 

AYP by safe-harbor. If any of the groups that failed to meet the status AMO also failed safe-harbor, 

then, in most of the pilot states, the number of non-proficient students who are on-track to proficiency 

per the growth model are added to the number of proficient students and the percentage proficient or 

on-track compared to the AMO. If one or more groups meets or exceeds the AMO because of the 

addition of the on-track students, the school is classified as making AYP by growth.
Evaluation of the Growth Model Pilot Project 10

Delaware adopted a different order, applying growth first and then applying status and safeharbor only to schools and subgroups that did not meet AYP under the growth criteria. This 

procedure thus classified schools as making AYP by growth even if they also would have made 

AYP by status or safe-harbor criteria. In contrast, the schools identified as making AYP by 

growth in the other states were all cases in which AYP was not met by status or safe-harbor.

Types of Growth Models Implemented in the Pilot

The states approved for the pilot study proposed to employ either a status-augmented-withgrowth or (in Delaware) a growth-augmented-with-status method of determining AYP, but the 

models used by the pilot states varied in how they established growth targets that define whether 

individual students were “on-track” to reach proficiency in the allotted time frame. The pilot 

states used three basic types of growth models: the transition matrix model, the trajectory

model, and the projection model.

16 

Transition Matrix. 

An illustration of the transition matrix model used in Iowa is shown in Exhibit 5. Students who 

scored below the proficiency threshold in year 1 (the first three rows) were classified as “ontrack” to proficiency if their test performances improved enough in year 2 to move up at least 

one level. All students who were proficient or advanced in year 1 and who were proficient or 

advanced in year 2 were classified as on-track. For the purpose of determining AYP, the Iowa

model counted students who were on-track toward proficiency as being fully proficient. All 

students in the gray-shaded cells are not on-track and do not count as proficient for AYP 

purposes.

This type of model evaluates student progress from year to year in terms of a 

small set of discrete performance levels. The levels are defined in general terms that are applied 

to all grades (e.g., below proficient, proficient, advanced). Student growth is indexed by 

movement (“transitions”) from lower to higher categories. Delaware and Iowa used models of 

this type.

Exhibit 5

Illustration of How the Iowa Transition Matrix Model Classifies Students

Year 1 

Performance Level

Year 2 Performance Level

Weak Low Marginal High Marginal Proficient Advanced

Weak Off-track On-track On-track On-track On-track

Low marginal Off-track Off-track On-track On-track On-track

High marginal Off-track Off-track Off-track On-track On-track

Proficient Off-track Off-track Off-track On-track On-track

Advanced Off-track Off-track Off-track On-track On-track

 16 See the CCSSO’s Implementer’s Guide to Growth Models for an alternative, more extensive, typology of growth 

models (retrieved May 2008 from http://www.ccsso.org/content/pdfs/IGG%20Final%20AP.pdf).
Evaluation of the Growth Model Pilot Project 11

Delaware used a somewhat different type of transition matrix model; it is illustrated in Exhibit 6. 

It used a point system that gave a maximum of 300 points to all students who attained 

proficiency in year 2 but gave only partial credit to students who scored below proficiency in 

year 1 and who made gains but did not reach the proficiency standard in year 2. Students who 

scored below the proficiency threshold in year 1 and did not move up to a sufficiently higher 

level in year 2 (the gray-shaded cells) were assigned no points for determining AYP. Points 

were summed within subgroups and divided by the number of students in each subgroup to 

calculate an “average growth value” which was compared to annual growth targets. 

Exhibit 6

Illustration of How the Delaware Transition Matrix Model 

Assigns Points to Students’ Gains

Year 1 Performance Level

Year 2 Performance Level

PL 1A PL 1B PL 2A PL 2B PL 3, 4, and 5

PL 1A: lowest level below 

proficiency 0 150 225 250 300

PL 1B 0 0 175 225 300

PL 2A 0 0 0 200 300

PL 2B: highest level below 

proficiency 0 0 0 0 300

PL 3, 4, and 5:

proficient or higher 0 0 0 0 300

Trajectory. 

An example of a trajectory model is shown in Exhibit 7. The Y axis represents student 

achievement on a vertically aligned scale, and the X axis represents successive grades. The large 

solid dots indicate the level of achievement necessary to be considered proficient at each time 

point (“p1” for proficiency at time 1, “p2” for proficiency at time 2, and “p3” for proficiency at 

time 3). The hollow dots represent a student’s actual achievement at time points 1 and 2, marked 

“s1” and “s2” respectively. 

This type of model uses the gap between a student’s baseline test score and a 

performance standard several years out to calculate the amount of growth that student must attain

to become proficient. This performance “trajectory” is divided up into annual growth targets 

that, taken together, put the student on-track to reach a grade-level proficiency cut score within 

the allotted years. Alaska, Arizona, Arkansas, Florida, and North Carolina used trajectory 

models.
Evaluation of the Growth Model Pilot Project 12

Exhibit 7

Illustration of How Trajectory Models Set Targets for 

Students Scoring Below Proficiency Thresholds

Exhibit reads: Under a trajectory model with a fifth-grade time limit, a student 

with a third grade score s1 below the proficiency cut score p1 must score at 

least as high as s2 in the fourth grade in order to be on-track to proficiency by 

fifth grade.

At grade 3, the student is not considered proficient because his or her achievement (“s1”) is 

lower than the proficiency cut score at that time (“p1”). Under a trajectory model, each student 

has a growth target that must be met to be considered “on-track” and count as proficient. In the 

illustration above, this is done by drawing a line from the student’s achievement at grade 3 (“s1”) 

to the proficiency point for grade 5 (“p3”). This line intersects point “s2” at grade 4, indicating 

that “s2” is what the student must achieve in grade 4 to be considered “on-track” to proficiency. 

The level of achievement targeted by the trajectory model is lower than what is expected under 

the status model. Under the status model, a student must have a level of achievement at or above 

point “p2.” 

This illustration shows that trajectory models allow students to score below the proficiency 

thresholds between the year they first miss proficiency (“s1”) and the year when they must be 

proficient under status criteria (“p3”). Because expected growth is based on a trajectory from the 

student’s initial achievement and the target year, trajectory models distribute the growth required 

to meet status expectations among the intervening years. While this illustration shows a linear 

set of proficiency cut scores and a corresponding linear trajectory for being on-track, some states, 
Evaluation of the Growth Model Pilot Project 13

such as Arkansas, have a nonlinear path of cut scores and use a nonlinear method to calculate 

growth targets. 

Projection.

Exhibit 8 illustrates a simple projection model. The projection model determines a projected 

level of achievement for a future time point for each student. Depending on the specifics of a 

state’s model, students who do not make the proficiency threshold under a status model at grade 

5 may still be counted as proficient if the projection model predicts that they will make 

proficiency by grade 6. Like the trajectory model illustration, the Y axis represents student 

achievement on some vertical scale. The X axis represents grades. Again, the solid dots 

represent grade-level proficiency thresholds at grades 3 (p1) through 6 (p4), and the hollow dots 

represent a student’s achievement over consecutive grades. The solid line represents the 

projection equation based on the current and past data for that student (points “s1” through “s3”). 

Though illustrated here with data for just one student, both the Ohio and Tennessee models 

estimate this projection equation from a regression analysis of students from a standard-setting 

cohort that most recently completed the eighth grade, and then use the regression coefficients to 

predict current students’ future scores based on their prior scores. The dotted line is the 

extension of that projection equation to a future time point specified by the state’s growth model. 

 This type of model uses current and past test scores to statistically predict 

performance several years ahead based on how all students in the state or school with similar 

patterns of scores generally perform. Such “projections” use multiple test scores from a 

reference cohort of students to estimate prediction equations using multiple regression 

techniques. For example, the eighth-grade cohort of students in 2007–08 could be the reference 

cohort, and the regression would estimate equations relating all of the past test scores for these 

students to their grade 8 reading and mathematics scores. These equations are then used to 

predict how each student in younger cohorts (e.g., the seventh-grade cohort in 2007–08) will 

score at the end of the eighth grade, given their current and prior test scores. If that predicted 

score is equal to or greater than the proficiency cut point, the student is classified as on-track to 

proficiency. Ohio and Tennessee are using projection models.

Under a status model, a student who reached the level of “s3” would not be considered proficient 

at grade 5, because his or her score is lower than that of point “p3”. However, under a growth 

model using projections based on student data, that student would be considered “on-track”

because his or her projected score meets or exceeds the proficiency cut score at grade 6 (the 

dotted line is above “p4” in the figure). 
Evaluation of the Growth Model Pilot Project 14

Exhibit 8

Illustration of How a Projection Model Predicts Future Achievement Based on Prior 

Scores

Exhibit reads: Under the projection model, a regression line is estimated on 

the basis of past scores s1, s2, and s3 to predict sixth-grade achievement.

How Growth Model Results Are Used for School AYP Determinations

The pilot states used the student-level indicators of on-track-to-proficiency generated by the 

GMPP growth models to augment the status indicators of proficiency in order to make AYP 

determinations for their schools. Eight of the nine states with data available for 2007–08 

followed what is referred to in the rest of the report as a “status-plus-growth” methodology. This 

involved using the standard status model first to assess whether the school made AYP. If the 

school did not make AYP by status, safe-harbor criteria (i.e., a 10 percent reduction in the 

number of non-proficient students from the prior to the current year) were applied. If the school 

still did not make AYP, only then were the growth model results used. 

An important aspect of the AYP determinations is that they are rooted in reading or language arts 

and mathematics achievement outcomes for all grade-eligible students in the school and also for 

each official reporting subgroup of students. Before the achievement scores are evaluated, an 

initial screening is done to assess whether the enrollment numbers and participation rates are 

sufficient. In terms of enrollment numbers, AYP determinations must be made for all schools 

regardless of size. However, each state identifies a minimum number of students that must be 
Evaluation of the Growth Model Pilot Project 15

enrolled in order for a subgroup to be included in AYP calculations; in most states, this 

“minimum n” is 30 or 40 students. If a school does not enroll enough students in a subgroup, the 

students in that subgroup are counted in the “all students” category, which every school must 

define. For the enrolled students, the federal law requires that a minimum of 95 percent must 

take each test, and this rate must be realized for all students and for each subgroup reaching the 

minimum n size for participation.

17

In the eight status-plus-growth pilot states, a school was designated as “made AYP by status” if 

all students and all subgroups had sufficient numbers of students scoring at or above the 

proficiency level in both subjects to meet their annual measurable objective (AMO, usually 

defined as a percentage of students at or above proficiency). If one or more subgroups failed to 

meet one or both AMOs, then the multiyear averaging and confidence interval methods 

described on p. 3 would be applied to the subgroup(s), still under the auspices of “status.” If one 

or more subgroups still did not meet an AMO, then the safe-harbor method was applied to the 

subgroup(s) and, if the group(s) then met the criterion of a 10 percent reduction in the number of 

non-proficient students, the school was designated as “made AYP by safe-harbor.” 

 If one or more eligible subgroup did not attain 95 percent 

participation, the school fails to make AYP regardless of test score results.

Only at the end of the status and safe-harbor tests were the growth model on-track indicators 

used in the status-plus-growth states. If one or more subgroups still did not pass after status and 

safe-harbor, six of the pilot states added the non-proficient students who were on-track according 

to the growth model to the proficient students and calculated the percent “proficient or on-track.”

In two states, the on-track indicators were used for all students in the subgroups that had not 

passed by status or safe-harbor. In either case, if that new percent met or exceeded the AMO, the 

group was classified as “made AYP by growth;” if all of the subgroups that failed to make AYP 

by status or safe-harbor passed when the on-track students were added in, then the school as a 

whole was classified as “made AYP by growth.” In the states that assess AMOs by adding ontrack students to those already proficient, it is thus possible for a schoolwide designation of 

“made AYP by growth” to result from just one non-proficient student being on-track to raise the 

percentage of “proficient plus on-track” students enough to meet the AMO. In general, then, 

“making AYP under growth” does not necessarily mean that all or even most students in the 

school are making sufficient learning gains to be counted as on-track to reach or (if already 

reached) maintain proficiency.

In contrast to the other eight pilot states, Delaware used a “growth-plus-status” method for 

determining AYP. As discussed more fully in Chapter II and Appendix B, Delaware used a 

point system whereby all students scoring proficient or higher earned a maximum number of 

points and students scoring below the proficiency cut score earned some fraction of that 

maximum depending on how much they improved from the prior year. If the average of the 

proficient (full credit) and non-proficient (partial credit) students’ points met or exceeded an 

annual target AMO, then the subgroup was classified as making AYP by growth in that subject. 

If all students and subgroups met the AMO, then the school was classified as making AYP by 

growth. If one or more subgroup did not meet the AMO, then multiyear averaging and 

 17 States are allowed to set their minimum n and the numbers range from no minimum to 100 in California.
Evaluation of the Growth Model Pilot Project 16

confidence intervals were applied to their point averages to see if they overlapped with the target 

AMO. If not, then safe-harbor was applied. 

Exhibit 9 provides an overview of each pilot state’s growth model organized by the type of 

model used and provides brief model summaries along dimensions that may affect final AYP 

outcomes. These include which grades receive growth calculations, how many years of growth 

are allowed, which tests are used to measure achievement, how students are identified as ontrack per the growth model, and how the on-track data are used for school AYP determinations. 

More detailed information about the technical features of each model can be found in the growth 

model summaries in Chapter II and also in Appendix B.
Evaluation of the Growth Model Pilot Program 17

Exhibit 9

Overview of Growth Models Approved for the GMPP in the 2007–08 School Year

State

First Year 

Used

Grades 

Included

Minimum 

Subgroup 

Size

Years to 

Proficient

Achievement 

Measures Used

How is Student On-Track-to-Proficiency 

Determined?

How are On-Track 

Determinations Used for AYP 

Determinations?

Delaware 2006–07 3–10 40 N/A Scale scores for 

reading and math 

via the Delaware 

Student Testing 

Program

Transition matrix model: Four levels 

of “below the standard” are used to 

categorize non-proficient students. A 

point system awards full credit for 

proficiency and partial credit for 

movement to higher levels of below 

proficient.

Students’ points are averaged 

to make AYP determinations 

for each subgroup. If the 

subgroup’s average meets or 

exceeds the AMO, the 

subgroup makes AYP.

Iowa 2006–07 4–8 30 4 Scale scores based 

on biannual Iowa 

Test of Basic Skills 

(ITBS) math and 

reading results

Transition matrix model: Three 

categories of performance are used 

to classify non-proficient students. 

Non-proficient students who move to 

a higher category of non-proficiency 

are on-track.

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Alaska 2006–07 4–9 25 4 Adjusted scale 

scores for math and 

language arts 

(reading + writing) 

on the Standards 

Based Assessment 

(SBA) Test

Trajectory model: Annual growth 

targets for each student are based on 

the test score gains needed to 

maintain or reach proficiency in four 

years or by grade 10. Students who 

close the test score gap by the 

reciprocal of the years remaining 

(e.g., 1/4, 1/3, 1/2) are “on-track.”

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Arizona 2006–07 4–7 40 3 Regression-adjusted 

scale scores for 

math and reading on 

the Arizona 

Instrument to 

Measure Standards 

(AIMS) tests

Trajectory model: : Annual growth 

targets for each student are based on 

the test score gains needed to 

maintain or reach proficiency in three 

years or by grade 8. Students who 

close the test score gap by the 

reciprocal of the years remaining 

(e.g., 1/3, 1/2) are “on-track.”

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Exhibit 9 continued next page
Evaluation of the Growth Model Pilot Project 18

Exhibit 9 (continued from previous page)

State

First Year 

Used

Grades 

Included

Minimum 

Subgroup

Size

Years to 

Proficient

Achievement 

Measures Used

How is Student On-Track-to-Proficiency 

Determined?

How are On-Track 

Determinations Used for AYP 

Determinations?

Arkansas 2006–07 4–7 40 4 Scale scores for 

literacy and math 

on the Arkansas 

Benchmark Exams

Trajectory model: Annual growth 

targets for each student are based on 

the test score gains needed to 

maintain or reach proficiency in four

years or by grade 8. Students who 

close the test score gap by a gradespecific amount are “on-track.”

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Florida 2006–07 3–10 30 3 Developmental 

Scale Scores 

(DSS) for math and 

reading on the 

Florida 

Comprehensive 

Assessment Test 

Trajectory model: Growth targets are 

based on the gap between initial score 

and proficiency cut scores three years 

later. Non-proficient students who 

close the original gap by one-third 

(one-half for ninth-graders) annually 

are on-track.

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, the 

percentage on-track in the 

subgroup is compared to the 

AMO.

North 

Carolina

2005–06 3–7 40 4 Scale scores on fall 

third-grade pretest 

and annual North 

Carolina End-ofGrade Math and 

Reading Tests

Trajectory model: Growth targets are 

based on closing the gap between 

baseline test score and proficiency 

cutoff in four years or by grade 8. 

Students who close the test score gap 

by 1/4 each year are on-track.

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Ohio 2007–08 4−7 30 3 Scores for reading 

and math on the 

Ohio Achievement 

Tests (plus the 

Ohio Proficiency 

Tests for the first 

cohort)

Projection model: Predicted scores for 

the grade beyond the school’s terminal 

grade or the grade three years later

are calculated for each student based 

on current and prior test scores; if 

predicted score plus two SE units is 

above the cut score for the target 

grade, student is on-track.

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, ontrack students are added to 

proficient students.

Tennessee 2005–06 4–8 45 4 Scale scores for 

math and 

reading/language 

arts from the 

Tennessee 

Comprehensive 

Assessment 

Program (TCAP)

Projection model: Predicted scores for

grade 8 are calculated for each 

student based on current and prior test 

scores; if predicted score is above the 

cut score for the target grade, student 

is on-track.

Status and safe-harbor applied 

first; if a subgroup does not 

make AYP with either, the 

percentage on-track in the

subgroup is compared to the 

AMO.
Evaluation of the Growth Model Pilot Project 19

Data Sources and Availability

Data employed for the analyses in this report were provided by the U.S. Department of 

Education and the pilot grantee states. The data from the Department were extracted from the 

EDFacts database. EDFacts is the main repository for school, district, and state data on ESEA

requirements related to making AYP determinations, as well as enrollment and demographic 

data. All states are required to submit standard data elements on all their districts and K–12 

schools each year to EDFacts. For the 2007–08 school year data that are the focus of this report, 

the standard reporting variable on whether the school made AYP was modified to collect 

information on whether each school made AYP because of the growth model. This variable was 

defined in EDFacts with a set of mutually exclusive categories: “made AYP by regular 

determination,” “made AYP by growth,” or “did not make AYP.” 

The “made AYP by regular determination” category included all methods of making AYP except 

by growth criteria. Regular determination included status criteria alone, status with confidence 

intervals, status with multiyear averaging, and safe-harbor (see p. 3 for definitions of these 

terms). Of these, only safe-harbor was reported separately. The standard EDFacts reporting 

variable for AMO results in 2006–07 included the mutually exclusive categories of “met by 

status,” “met by safe-harbor,” “exempt by minimum n,” and “did not meet” for each subject area. 

A new reporting variable for “met by growth” was added to subgroup AMO results in 2007–08. 

For purposes of this report, schools were classified as “made AYP by safe-harbor” if the school 

as a whole was identified as “made AYP by regular determination” and at least one subgroup

was classified as “met by safe-harbor” for either the reading or mathematics AMO.

The nine GMPP states were additionally required as a condition for continued participation in the 

pilot to collect and make available for evaluation purposes student-level data on growth model 

outcomes. The data obtained directly from the pilot states consisted of scale scores and 

proficiency designations in reading or language arts and mathematics for each student in the 

grades involved in the GMPP. Of particular importance to this study were the proficiency 

designations, for these included the indicator of whether the student was “on-track” to achieve 

proficiency within the time frame specified by the approved growth model. Additional data 

elements included various background characteristics including school identifier codes, grade 

level of current enrollment, and ESEA subgroup memberships.

Eight of the nine states had provided both the EDFacts and student-level data required to address 

the two study questions which are the focus of this final report. The other state (Delaware) 

reported GMPP results to the EDFacts archive but, because of the unique way in which it used 

the growth model data, did not provide indicators of which schools classified as having made 

AYP by growth also would have made AYP if the growth data were not used. However, the 

Delaware Department of Education was able to provide that information separately from the 

EDFacts data.
Evaluation of the Growth Model Pilot Project 20

Calculation of Hypothetical “Growth-Only” Outcomes

In addition to the EDFacts-reported outcomes for subgroups and schools, the student growthmodel results are used in Chapter IV to calculate percentages of students who were on-track to 

reach proficiency. These “growth-only” indicators are used to address hypothetical “what if”

questions that go beyond the basic descriptive aims of Chapters I–III. 

As noted above, states had the options of making AYP determinations using (a) only growth 

results and (b) growth criteria before status and safe-harbor criteria. While these options were 

generally not exercised, it is possible to estimate what would happen if they had been. That 

information may prove useful to states in the process of developing growth models, as well as 

current pilot states contemplating changes to their models. Toward that end, the student 

indicators of on-track-to-proficiency are used to calculate hypothetical variants of AMO 

outcomes based on the percentages of on-track students in reading and mathematics for each 

ESEA subgroup in each school. These percentages are compared to the state’s AMOs to assess 

the extent to which schools currently making AYP by status and safe-harbor would be able to 

reach their AMOs using growth-only percentages of students on-track to proficiency.

Data Limitations

The data described above have a number of limitations that are important to note at the outset 

and which will be reiterated at various points in this report. One limitation is that the mutually 

exclusive AYP categories reported by EDFacts made it impossible to determine the extent to 

which schools that made AYP under status and safe-harbor would also have made AYP if only

growth criteria were considered. The student-level on-track-to-proficiency indicators described 

in the preceding subsection can be used to calculate growth-based AMO determinations for each 

ESEA reporting group and for the school as a whole. However, it should be emphasized that 

these growth-based AMO determinations are based solely on information regarding students’

reading or language arts and mathematics performances and do not necessarily indicate whether 

a school would make AYP under a growth-only system. Under ESEA, schools must also meet 

additional conditions to make AYP. These conditions include the requirements that the school 

(a) meet or exceed a minimum level on an “other academic indicator,” typically average daily 

attendance for elementary and middle schools and graduation rates for high schools; and (b) 

realize at least a 95 percent participation rate on the annual assessments.

A second limitation is that we were unable to consider these criteria in our calculations because 

the requisite data were not available. For example, the “other academic indicator data” are 

school-level determinations that were not provided in the student files and were unevenly 

reported in EDFacts. The student files from some states also did not include the information 

needed to calculate participation rates.18

 18 Participation rates are calculated on the basis of students enrolled for the “full academic year” (FAY), but FAY 

and non-FAY students were not distinguished in some of the state data files. 

This means that some schools meeting the AMO on the 
Evaluation of the Growth Model Pilot Project 21

achievement outcomes using the growth criteria identified in this report may have officially 

missed AYP on the additional criteria.19

Given the potential discrepancies between the growth-based AMO designations and the official 

ESEA designations, the results presented in Chapter IV of this report should be regarded as 

suggestive rather than definitive. With that caveat, these comparisons allow an assessment of the 

extent to which schools making AYP by status or safe-harbor criteria might also have made AYP 

if only growth criteria were used. 

 

 19 The number of such “false positives” is probably small. National data from 2003–04 show that about 3 percent of 

all schools did not make AYP solely because of not making acceptable levels on the other academic indicator or low 

(below 95 percent) participation rates for the achievement testing (U.S. Department of Education, 2007, p. 43.)
Evaluation of the Growth Model Pilot Project 22
Evaluation of the Growth Model Pilot Project 23

II. Effects of Growth Models

Implemented Under the GMPP on School AYP Determinations

This chapter presents overviews of the growth models implemented in the nine pilot states, 

focusing on the methods they employed to (1) set expectations, (2) measure growth, and (3) 

incorporate these growth results into their AYP determinations. The nine states provided data on 

the results of their pilot growth models and, following the overview of each state’s model, we 

present evidence of the impact of their growth model on AYP outcomes. 

The model summaries provided here are based on a review of extant documentation, including 

the approved GMPP proposals, decision letters from the U.S. Department of Education, and 

correspondence with the Department concerning feedback and suggested revisions from the peer 

review panels; growth model descriptions found in state accountability workbooks, school report 

cards, and other online technical documentation; and edits by state officials to draft model 

summaries provided to each pilot state at a December 2008 summit meeting in Washington, D.C. 

Any inconsistencies in model summaries were then resolved through e-mail correspondence and 

follow-up phone calls with state officials over the course of the evaluation. More detailed 

technical summaries of the states’ growth models are included in Appendix B.

The data analyses address the research question “How many schools made AYP under the 

growth model that would not have made AYP under the ESEA status model?” For the eight 

states that participated in the GMPP for at least two years (Ohio joined the pilot project in 

2007–08), we present data on the growth model outcomes in both the 2006–07 and 2007–08 

school years. The data from both years are drawn primarily from the school-level reports from 

each state to the federal EDFacts system maintained by the U.S. Department of Education. 

State Growth Models and Their Effects on Schoolwide AYP Results

Alaska’s Growth Model

Alaska was formally accepted into the Growth Model Pilot Project on July 3, 2007. Alaska’s 

growth model includes fourth- through ninth-graders only, with all other students expected to be 

proficient under the status model criteria. The state uses a trajectory model that defines targets 

for each student at each grade based on the student’s scores on the Standards Based Assessment 

(SBA) tests for mathematics and language arts. The SBA test was first given in the 2004–05 

school year to students in grades 3–9. The SBA is scaled such that students in every grade must 

score 300 or above to be considered proficient in math and a combined reading and writing score 

of 600 or above to be considered proficient in language arts.20

 20 Alaska estimates a “true score” for the SBA by adjusting for the annual reliability of the tests. This has the effect 

of raising scores that are below average. The estimated true score is calculated by subtracting an estimated “reliable 

deviation” of the student’s observed score from the statewide average for students in the same grade.

 
Evaluation of the Growth Model Pilot Project 24

Starting in the 2006–07 school year, fourth- through sixth-grade students scoring below the 

proficiency cutoff are counted as proficient for AYP purposes if they are on-track to reach 

proficiency by the seventh grade, and eighth- and ninth-graders are counted as proficient if they 

were on-track to reach proficiency by the tenth grade. Students are always only classified by the 

status model in grades 3 and 10. Annual growth targets for all fourth-, fifth-, sixth-, seventh-, 

eighth-, and ninth-graders who scored below proficiency are set by dividing the difference 

between the student’s “baseline” test scores and the proficiency cutoffs (300 in math and 600 in 

language arts) by the number of years allowed. Students scoring below the proficiency cut score 

are classified as on-track to proficiency if they (1) meet or exceed their target score defined by 

the growth model, and (2) do not score lower than they did in the prior year.

The Alaska growth model only applies to students who score below proficient on the SBA in two 

or more consecutive years within the span of grades 3 through 8. The baseline scores used in the 

growth trajectory calculations are defined as those from the first year the student scored below 

the proficiency cutoff. For students in grades 4 through 6 in the 2007–08 school year, this could 

go back to the third grade (i.e., to 2004–05 when the SBA was first administered). Thus for 

fourth-grade students who scored below proficiency in both the third and fourth grades, the 

baseline is the third-grade score. For fourth-grade students who were at or above the proficiency 

cut score in third grade but below proficiency in fourth grade, the baseline is the fourth-grade 

score. For fifth-grade students who scored below proficiency in the third and fourth grades, their 

third-grade scores would also be the baselines used to define their trajectories to proficiency by 

grade 7. The baseline for students in grades 8 and 9 who score below proficiency can go back to 

seventh grade. Students who change schools or districts carry their baseline score with them.

A third-grade student who scored below proficiency has four years of allowable growth and must 

make up one-quarter of the gap between his or her third-grade score and the seventh-grade

proficiency cut score in the first year, one-half of this proficiency gap by the end of the second 

year, three-fourths of the gap by the end of the third year, and must score at or above the cut

score in the fourth year in order to be counted as proficient for AYP determinations. Similarly, a 

seventh-grader scoring below proficiency must close the gap by one-third in grade 8 and by twothirds by the end of grade 9.21

The Alaska growth model uses a “status-plus-growth” method for determining whether reporting 

groups meet the AMOs for language arts and mathematics. That is, first, the reporting group is 

assessed in terms of the percentage of students who scored at or above proficiency. If that 

percentage met the AMO, the group was classified as “met by status.” If the percentage did not 

meet the AMO but the percentage not proficient was 10 percent or more less than the percentage 

not proficient in the prior year, the group was classified as “met by safe-harbor.” If the group did 

 A student’s baseline score is not reset in terms of the prior year 

score as long as the student continues to score below the proficiency cut score. However, the 

student must score at least as high as he or she did in the prior year as well as achieve the 

requisite gap reductions calculated from the baseline score in order to be counted as on-track to 

proficiency.

 21 Note that a ninth-grader must close the proficiency gap by 100 percent by the tenth grade, meaning that the 

growth model is equivalent to the status model for all ninth-graders who score below proficiency.
Evaluation of the Growth Model Pilot Project 25

not meet the AMO by status or pass by safe-harbor, then the number of below-proficient students 

who are on-track to proficiency was added to the number of proficient students and this sum was 

divided by the total number of students in the reporting group who took the test. If this 

percentage met the AMO, the group was classified as “met by growth.” If one or more reporting 

groups met by growth and the other AYP criteria noted below were also met, the school as a 

whole was classified as “made AYP by growth.”

Alaska’s ESEA accountability rules require a reporting subgroup to have more than 25 students 

before it is included in school AYP determinations. Subgroups larger than 40 students count 

only if 95 percent of them participate in testing, while smaller subgroups require that no more 

than two students fail to participate. Subgroups that meet the AMO within a 99 percent 

confidence interval, which varies by subgroup size, are determined to have made AYP.

The AYP results for Alaska in the 2006–07 and 2007–08 school years are shown in Exhibit 10. 

Of the 292 schools that met AYP requirements (222 by status plus 70 by safe-harbor) for that 

year, representing 59 percent of all eligible schools, no school made AYP via the state’s pilot 

growth model. In 2006–07, 323 schools met AYP requirements, representing 66 percent of all 

eligible schools, and again no school made AYP via the state’s pilot growth model.

Exhibit 10

Alaska School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 323 222 66% 45%

Met with Safe-Harbor 0 70 0% 14%

Met with Growth 0 0 0% 0%

Not Met 169 203 34% 41%

All Eligible Schools 492 495 100% 100%

Exhibit reads: For Alaska’s schools overall, 222 met AYP under status in 2007–08, which 

was 45 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Arizona’s Growth Model

Arizona’s participation in GMPP was approved on July 3, 2007, for use beginning in the 

2006–07 school year. The Arizona growth model includes students in grades 4 through 7 and 

uses scores on the Arizona Instrument to Measure Standards (AIMS) tests for reading and 

mathematics, which are vertically scaled for grades 3 through 8. Proficiency cut scores are set 

for each grade, and a student scoring below the cut score for either reading or math has three 

years or by eighth grade, whichever comes first, to reach the cut score. 

The state uses a trajectory model that sets growth targets by dividing the difference between 

initial score and proficiency cut score three grades later into equal parts. In order to be counted 
Evaluation of the Growth Model Pilot Project 26

as “on-track” for AYP purposes, a student in grades 3 through 5 must make up one-third of the 

shortfall in the first year, two-thirds of the shortfall by the end of the second year, and be 

proficient by the end of the third year. A sixth-grader must cover half of the shortfall in each of 

the two years of eligibility remaining.22

For the purposes of its growth model, Arizona calculates predicted AIMS scores by regressing 

current year’s scores on previous year’s scores and unique identifiers of the schools in which the 

students are enrolled. The student’s actual previous year score and school identifier are 

multiplied by the respective regression coefficients to calculate a predicted score. These 

predicted scores are then subject to a 95 percent confidence interval, and the lower bound of that 

confidence interval is used as the student’s score for comparing to his or her growth target. This 

regression-based adjustment procedure is done in order to correct for improvement that might be 

due to chance. This adjusting process has the effect of discounting high gains among lowerperforming students and discounting declines among higher-performing students, because those 

patterns are anomalous with respect to the regression equation.

Students who leave the Arizona school system before 

reaching proficiency have new growth targets set upon their return.

The state also uses a 99 percent confidence interval around annual AMOs for AYP 

determinations under the status model.

23

Arizona applied the results of the growth model after applying status and safe-harbor criteria. 

Growth model results were only applied to ESEA reporting groups that did not make AYP by 

status or safe-harbor. For those groups the number of students identified as “on-track to 

proficiency” was added to the number of proficient students and that sum was divided by the 

number of test-takers in the reporting group. That percentage was then compared to the AMO. 

If the school met all AYP criteria and had one or more reporting groups meeting the AMO 

because of the addition of on-track students, the school as a whole was classified in EDFacts as 

making AYP by growth.

ESEA subgroups are counted in such determinations 

only if they have at least 40 students and if no fewer than 95 percent of these students participate 

in annual testing. Students who have estimated scores above the proficiency cutoff but who also 

fail to meet their growth targets continue to be counted as proficient for AYP purposes.

EDFacts data for the 2007–08 school year indicate that Arizona’s growth model resulted in only 

eight schools making AYP that would have missed if only status and safe-harbor were applied 

(Exhibit 11). Of the other 1,117 schools that met AYP requirements, 97 made AYP using the 

safe-harbor criteria, and 1,020 schools did so using status criteria alone. Data for the 2006–07 

school year indicate that just one school made AYP via the growth model, while no schools 

made AYP via safe-harbor.

 22 Note that a seventh-grader must close the proficiency gap by 100 percent in the first year, meaning that the growth 

model is equivalent to the status model in this grade.

23 Confidence intervals around the AMO (as opposed to around the percent proficient) are calculated using the 

number of full academic year students in the reporting group as the basis for the estimate. If the percent proficient in 

the group meets or exceeds the lower bound of this confidence interval, the group is classified as meeting the AMO.
Evaluation of the Growth Model Pilot Project 27

Exhibit 11

Arizona School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 1,096 1,020 77% 68%

Met with Safe-harbor 0 97 0% 6%

Met with Growth 1 8 <1% 1%

Not Met 333 371 23% 25%

All Eligible Schools 1,430 1,496 100% 100%

Exhibit reads: For Arizona’s schools overall in 2007–08, 1,020 met AYP under status, which 

was 68 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Arkansas’ Growth Model

Arkansas received approval to implement its proposed growth model in the 2006–07 school year. 

Arkansas uses a trajectory model that calculates growth for students in grades 4 through 7 using 

results of the Arkansas Benchmark Exams for mathematics and literacy, which are administered 

in grades 3 through 8 (plus grade 11 for literacy). Proficiency levels for these vertically scaled 

exams are set for each grade, and growth targets are based on the annual exam score increment 

needed to reach the proficiency standard in eighth grade. The annual increment that a student 

must attain in order to be classified as on-track to proficiency is calculated using grade-specific 

growth target multipliers of 0.295 in fourth grade, 0.319 in fifth grade, 0.385 in sixth grade, and 

0.542 in seventh grade. These multipliers indicate the proportion of the total difference between 

the eighth-grade standard (a score of 700) and the student’s current score that the student must 

gain over the next year in order to be on-track for eighth-grade proficiency. They contrast with 

the multipliers in other trajectory states (which all represent fractions of whole years—e.g., onefourth, one-third, one-half) because Arkansas Benchmark Exams are not scaled to have a linear 

progression from one year’s cut point to the next (see Appendix B for more detail and 

illustrations). 

An important feature of this model is that it resets the growth target every year rather than setting 

a series of annual targets based on the first below-proficient exam score. For example, a thirdgrader who scores a 480 on the Arkansas Benchmark Exams is 20 points below the standard for 

that grade. Since the eighth-grade standard is 700, the student must gain 65 points [= (700–480) 

* 0.295] to be counted as on-track to proficiency. If the student scores a 539 in fourth grade 

(again 20 points below the standard), he or she fails to reach this threshold and now needs to gain 

51 points [=(700–539) * 0.319] in fifth grade to be counted as on-track to proficiency. Arkansas 

students who make sufficient growth are counted as proficient for AYP determinations. 

Arkansas applies confidence intervals to the grade-specific AMOs. Most of the other states in 

the GMPP apply confidence intervals to the percentages of proficient students rather than to the 
Evaluation of the Growth Model Pilot Project 28

AMO. The confidence interval is 95 percent and is applied by using the number of students 

enrolled in the school’s tested grades for each reporting subgroup. 

The data reported to EDFacts by Arkansas for the 2007–08 school year show that 15 percent of 

the schools made AYP according to the status model, 41 percent made AYP under safe-harbor

provisions, and 6 percent made AYP under the growth model (Exhibit 12). These 6 percent 

represent schools that would not have made AYP had the GMPP not been available. In 2006–07, 

8 percent of Arkansas schools made AYP via growth, while 24 percent made AYP under the 

status model and 32 percent made AYP using safe-harbor provisions.

Exhibit 12

Arkansas School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 212 131 24% 15%

Met with Safe-Harbor 287 369 32% 41%

Met with Growth 69 52 8% 6%

Not Met 326 338 36% 38%

All Eligible Schools 894 890 100% 100%

Exhibit reads: For Arkansas’ schools overall, 131 met AYP under status in 2007–08, which was 

15 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

The percentages of Arkansas schools making AYP with safe-harbor were much higher than in 

any of the other GMPP states in both years. It is not clear why this occurred. 

Delaware’s Growth Model

Delaware also received approval to use its proposed growth model starting with the 2006–07 

school year. Delaware’s model includes students in grades 3 through 10 and does not limit the 

number of years students have to make proficiency. The Delaware Student Testing Program 

(DSTP) yields vertically aligned scale scores for mathematics and language arts (a combination 

of reading and writing). Mathematics and reading tests are administered starting in grade 2 and 

growth can be assessed in grade 3 and above, but the writing component of the language arts 

scores does not start until grade 3. 

The Delaware model is an example of the transition matrix type. Delaware uses a “value table”

method for AYP determinations that assigns points for students depending on the type and extent 

of changes between the performance levels (see Exhibit 6). Points in the value table increase 

with the level of proficiency, so that a student scoring at the bottom receives fewer points for 

moving up one level (150) than a student moving from that level to the next (175). Conversely, 

students who move up two levels are awarded more points (225) than a student who moves to the 

same level in one step (175 or 200). All students who surpass their grade level proficiency cut
Evaluation of the Growth Model Pilot Project 29

score receive 300 points regardless of how high they score or whether their underlying scores 

actually declined from the prior year. AYP is determined by calculating the average points for 

the school and its subgroups in the two required subject areas and comparing these averages with 

the state’s AMO levels.

Delaware was unique in the pilot project in that it combined results of the status and growth 

models, and classified all schools that made AYP using the combined results as “met AYP by 

growth” even if they would have met AYP by status alone. The other states employed the statusplus-growth model shown in Exhibit 4. Delaware’s arrangement made it impossible to 

determine from EDFacts whether the schools listed as making AYP under growth also made 

AYP under status.

Drawing on the state reports in EDFacts, 87 of 183 schools in Delaware made AYP under the 

growth model in the 2007–08 school year (Exhibit 13). As a result of using the procedure of 

employing the growth model results before applying the status and safe-harbor provisions, the 

percentage of Delaware schools reported in EDFacts as making AYP under growth (48 percent) 

is much higher than most of the other pilot states. Many of these 87 schools would have made

AYP under status or safe-harbor, had those criteria been applied before (as was done in the other 

GMPP states) the growth criteria. In 2006–07, a similar 46 percent of the schools made AYP 

under growth before applying status and safe-harbor.

Exhibit 13

Delaware School AYP Determinations With Growth Model Results Augmented With 

Status and Safe-Harbor Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 21 41 12% 22%

Met with Safe-Harbor 19 0 11% 0

Met with Growth 83 87 46% 48%

Not Met 57 55 31% 30%

All Eligible Schools 180 183 100% 100%

Exhibit reads: For Delaware’s schools overall, 41 met AYP under status in 2007–08, which was 

22 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

The EDFacts reporting on school AYP outcomes used mutually exclusive categories of “by 

status” and “by growth” and it was thus not possible to use those data to address the question of 

how many Delaware schools that made AYP under their growth model would have missed AYP 

under the status model or by safe-harbor. However, data provided by the state directly to the 

evaluation team indicate that only five schools, or 3 percent of the schools in the state, made 

AYP because of the growth model provisions in both the 2007–08 and 2006–07 school years
Evaluation of the Growth Model Pilot Project 30

(Exhibit 14).24

Exhibit 14

Delaware School AYP Determinations With Status and Safe-Harbor 

Results Augmented With Growth Model Results, 2006–07 and 2007–08

 These were the only schools that were reclassified from not making AYP to 

making AYP as a direct result of Delaware’s participation in the GMPP. 

School AYP under Statusplus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 101 123 55% 67%

Met with Safe-harbor 22 0 12% 0%

Met with Growth 5 5 3% 3%

Not Met 57 55 31% 30%

All Eligible Schools 185 183 100% 100%

Exhibit reads: For Delaware’s schools overall, 123 met AYP under status in 2007–08, which 

was 67 percent of all eligible schools.

Source: Delaware Department of Education.

Florida’s Growth Model

Florida’s growth model proposal was approved for use starting with the 2006–07 school year. 

The growth model applies to students in grades 3 through 10 using the Developmental Scale 

Scores (DSS) from the Florida Comprehensive Assessment Tests (FCAT) for mathematics and 

reading. The state uses a trajectory model that bases growth targets on the score required for 

proficiency three years after the first year tested (normally, grade 3). To be counted as 

proficient, a student who was not proficient at the baseline must close the gap between his or her

baseline score and the proficiency cut score three grades later by one-third the first year and twothirds the second year.

25

Florida retains third-graders who score at or below level 1 (the lowest level) on the reading 

portion of the FCAT and who do not qualify for an exemption from this policy. The growth 

model incorporates such students by using their two third-grade scores on the FCAT DSS. 

Otherwise, only students in the fourth grade or later will have the two scores required to 

calculate growth. 

Students who continue to score below the cut score after three years 

start the process over and have new growth targets set.

For determining school AYP, Florida applies growth-model results for students only in ESEA

reporting groups that do not meet their AMOs by status or safe-harbor. An important feature of 

Florida’s accountability model is that, for groups not meeting their AMO, the growth-model data 

 24 These data were provided to NORC by the Delaware Department of Education’s Assessment and Accountability 

Branch. The data differed from the EDFacts reports in that both status and growth results were included for all 

schools and ESEA reporting groups. 

25 This means that a student can only make proficiency by growth for two years, because the gap must be fully 

closed in the third year; i.e., a student must meet or exceed the minimum proficiency score in year 3. A student first 

enrolled in grade 9 must close the gap by half to be counted as “on-track.”
Evaluation of the Growth Model Pilot Project 31

are used exclusively. That is, the number of non-proficient but on-track students is not added to 

the number of proficient students as is done in several other pilot states. Instead, the growth 

model results for both the proficient and non-proficient students are used. This means that 

students who are proficient but who are not on-track to maintain proficiency in three years are 

counted as “not proficient” for AYP purposes in these reporting groups. Groups not meeting 

AMO by status or safe-harbor must thus meet the AMO entirely on the basis of “on-track”

students. 

The AYP results for Florida in 2007–08 are shown in Exhibit 15. A total of 24 percent of 

Florida schools made AYP, with 16 percent making it under status, 4 percent by safe-harbor, and 

5 percent by growth. The previous year, 34 percent of Florida schools made AYP, including 5 

percent that made AYP via the pilot growth model. The overall percentage of schools that made 

AYP in Florida is much lower than the other GMPP states, but this may reflect higher standards 

for proficiency or higher AMOs rather than lower performance. 

Exhibit 15

Florida School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 770 516 24% 16%

Met with Safe-harbor 155 116 5% 4%

Met with Growth 149 153 5% 5%

Not Met 2,135 2,495 67% 76%

All Eligible Schools 3,209 3,280 100% 100%

Exhibit reads: For Florida’s schools overall, 516 met AYP under status in 2007–08, which was 

16 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Iowa’s Growth Model

Iowa’s growth model was approved for use with students in grades 3 through 8 beginning in the 

2006–07 school year. Iowa uses third-grade math and reading scores on the Iowa Test of Basic 

Skills (ITBS) as a baseline, so growth calculations begin in fourth grade. Iowa’s growth model 

is a type of transition matrix model (see Exhibit 5). To calculate growth, Iowa uses two 

categories of proficient (Intermediate and High) and three categories of below proficient (Weak, 

Low Marginal, and High Marginal). Category boundaries are set using national percentile ranks 

for each grade, with ITBS scale scores in the 40th percentile considered below proficient and 

10th percentile scores considered Weak. Non-proficient students can still make Adequate Yearly 

Growth (AYG) if they improve to a higher category of below proficient within four years of the 

first year tested.

An important feature of Iowa’s model is that students cannot fall back to a non-proficient 

category and still make AYG. This means that High Marginal students who decline to Low 

Marginal or Weak cannot make AYG simply by regaining High Marginal status, and that Low 
Evaluation of the Growth Model Pilot Project 32

Marginal students who decline to Weak must score in the High Marginal category to make 

adequate growth. Iowa counts all students making AYG as proficient for ESEA reporting groups 

that fail to make AYP when the status and safe-harbor provisions have been applied.

The 2007–08 AYP results for Iowa show that 2 percent of the schools made AYP with the 

growth model in the status-plus-growth framework (Exhibit 16). This was on top of the 62 

percent of schools making AYP under status and 4 percent via safe-harbor. In 2006–07, 11 

percent of Iowa schools made AYP using growth criteria, 75 percent made AYP under status,

and 10 percent made AYP via safe-harbor. 

Exhibit 16

Iowa School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 831 678 75% 62%

Met with Safe-harbor 105 43 10% 4%

Met with Growth 116 23 11% 2%

Not Met 52 354 5% 32%

All Eligible Schools 1,104 1,098 100% 100%

Exhibit reads: For Iowa’s schools overall, 678 met AYP under status in 2007–08, which was 62

percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

North Carolina’s Growth Model

North Carolina received approval on May 17, 2006, to use its growth model for the 2005–06 

school year. The state uses a trajectory model that calculates growth in grades 3 through 7 using 

vertically equated North Carolina end-of-grade tests for mathematics and reading. Third-graders 

in North Carolina have four years to grow to proficiency, because they take a test upon entering 

the third grade and a test at the end of the year. All other students take end-of-grade tests and 

thus have only three years or until eighth grade to become proficient, whichever comes first.

For students who take the third-grade pretest, growth targets are set by dividing the difference 

between the initial test score and the proficiency cut score for sixth grade into four equal parts. 

Thus third-grade students who make up one-fourth of the shortfall between the baseline score 

and sixth-grade standards by the end of third grade are considered to be “on-track” to 

proficiency. Students who enter the school system after third grade must close this gap more 

quickly, because they will have a three-year horizon before eighth grade, when all students are 

expected to make a proficient score. 

For AYP purposes, North Carolina applies growth-model results after status and safe-harbor

criteria. For reporting groups not meeting their AMO by status or safe-harbor, the number of 

non-proficient students who are on-track to proficiency per the growth model is added to the 
Evaluation of the Growth Model Pilot Project 33

number of proficient students in the group. A group that reaches the state’s AMO with the 

inclusion of “on-track” students is considered to have met the AMO by growth and the school as 

a whole is classified as having made AYP by growth. 

North Carolina applies a 95 percent confidence interval before using the growth model for AYP 

determinations and only includes subgroups with at least 40 full academic year (starting on 

Oct. 1 of the current school year) students. The standard 95 percent participation rule also 

applies to subgroups and schools and is based on the full set of students enrolled in the current 

school year. 

Exhibit 17 shows that the pilot growth model resulted in no more North Carolina schools making 

AYP in 2007–08 than would have made AYP through status (399) and safe-harbor (338) alone. 

More than two-thirds (69 percent) of schools in North Carolina did not meet AYP requirements, 

up from just over half (56 percent) of schools in 2006–07. Twelve schools also met their AYP 

requirements via the pilot growth model during that school year.

Exhibit 17

North Carolina School AYP Determinations With Status and Safe-Harbor Results 

Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 661 399 30% 17%

Met with Safe-harbor 308 338 14% 14%

Met with Growth 12 0 1% 0%

Not Met 1,226 1,612 56% 69%

All Eligible Schools 2,207 2,349 100% 100%

Exhibit reads: For North Carolina’s schools overall, 399 met AYP under status in 2007–08, 

which was 17 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Ohio’s Growth Model

Ohio was notified of its admission into the pilot program on August 15, 2007, subject to the 

condition that the state adopt a uniform minimum subgroup size. Such a resolution passed the 

Ohio General Assembly on April 23, 2008. Ohio’s approved plan uses a projection model for 

determining whether students are on-track to proficiency. It includes growth calculations for 

grades 3 through 7 using scaled scores for reading and mathematics on the Ohio Achievement 

Tests plus grade 10 scores on the Ohio Graduation Tests.26

 26 Ohio used (equated) 2004–05 data from the Ohio Proficiency Tests for eighth-graders in 2006–07 to give them at 

least three years of test results. 

Performance standards for each 

subject and grade are expressed as cut score points. Students have three years to make a 

proficient cut score, fewer if the student is expected to graduate from the current school before 

then. A student is considered to be on-track to proficiency if he or she is “projected” to meet the 
Evaluation of the Growth Model Pilot Project 34

cut score in three years or for the next grade beyond the current school’s terminal grade, 

whichever comes first. 

These projections of student achievement are derived from a complex statistical model designed 

to predict each student’s level based on data about his or her past performance and the school the 

student will most likely attend after his or her current school. The statistical model uses at least 

three and up to five years of all available test scores. The model uses regression analysis to 

calculate the relationships between the past scores and endpoint school and the target endpoint 

score for a “reference cohort” defined as the most recent cohort of students that completed the 

endpoint. Consistent with the GMPP core principles (see Exhibit 3), the model does not make 

any adjustments for demographic characteristics, meaning that students with differing 

socioeconomic or racial backgrounds but with the same prior academic achievement and likely 

next school will have the same projections. 

The regression coefficients estimated with the reference cohort are then used to calculate 

predicted endpoint scores for the students in the younger cohorts. Standard errors around the 

predictions are also estimated and two standard error units are added to each predicted score.27

Ohio uses the growth model results for AYP determinations only after applying status and safeharbor criteria. For subgroups that do not make AYP by status or safe-harbor, students scoring 

below proficiency but who are identified as on-track are counted the same as proficient students. 

If the proportion of proficient plus on-track students meets or exceeds the AMO, then the 

subgroup is classified as “making AYP by growth” in the federal reporting system. If all

subgroups make AYP by an available method and at least one subgroup makes AYP by growth, 

then the school as a whole is classified as making AYP by growth in the EDFacts system. 

This adjustment reduces the chance of misclassifying a student who is actually on-track as not 

on-track, but increases the chance of misclassifying a student who is actually not on-track as ontrack. If the adjusted prediction meets or exceeds the cut score for the target grade, the student is 

classified as on-track. An important feature of Ohio’s growth model is that projections are 

possible even if a student is missing prior achievement scores for some grades or subjects. Also, 

the model recalculates projections every year and calculates projections for students who score 

above proficiency. 

Ohio joined the GMPP in the 2007–08 school year and thus has only one year of results. Exhibit 

18 shows that the pilot growth model resulted in 983 more Ohio schools making AYP in 2007–

08 than would have made AYP through status (798) and safe-harbor (163) alone. About a third 

(34 percent) of schools in Ohio did not meet their AYP requirements by status, safe-harbor, or 

growth. Ohio had by far the largest reported percentage of schools making AYP through the 

growth model (34 percent) of all the pilot states; the states with the next highest percentages 

were Arkansas (6 percent) and Florida (5 percent). 

 27 Ohio Department of Education and SAS Institute, Inc., Conference Call, July 19, 2010.
Evaluation of the Growth Model Pilot Project 35

Exhibit 18

Ohio School AYP Determinations With Status and Safe-Harbor

Results Augmented With Growth Model Results, 2007–08

School AYP Under Status-PlusGrowth Model Number Percent

Met with Status 798 27%

Met with Safe-harbor 163 6%

Met with Growth 983 34%

Not Met 984 34%

All Eligible Schools 2,928 100%

Exhibit reads: For Ohio’s schools overall, 798 met AYP under status, which was 

27 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Tennessee’s Growth Model

Tennessee received approval on May 17, 2006, to use its growth model beginning in the 

2005–06 school year.28

For students past the fifth grade, proficiency cut scores are based on the TCAP high school 

graduation (or Gateway) assessments. As is the case with the Ohio growth model, the Tennessee

growth model is able to make projections even if a student is missing prior achievement scores 

for some grades or subjects. The Tennessee model also recalculates projections every year and 

calculates projections for students who score above proficiency. 

The state calculates growth for students in grades 4 through 8 using 

vertically aligned scores for mathematics and reading or language arts from the Tennessee 

Comprehensive Assessment Program (TCAP). Students are first tested in the third grade, and 

those who score below proficient are given until ninth grade to reach proficiency by growth. 

Tennessee’s projection model establishes growth targets for students by calculating a predicted 

ninth-grade achievement score for each student on the basis of his or her current and prior test 

scores. As in Ohio, predicted scores are calculated on the basis of regression coefficients 

estimated using data from a standard-setting cohort that has just completed the eighth grade. 

Unlike Ohio, however, Tennessee does not adjust the predicted scores with estimates of standard 

errors. Students who score below proficiency but who are predicted to be above the TCAP cut 

scores within three years or by grade 9, whichever comes first, are counted as proficient for AYP 

growth determinations.

For AYP purposes, the growth-model results are applied only to ESEA reporting groups that did 

not meet their AMOs by status or safe-harbor. As in Florida, Tennessee uses growth-model 

results for all students in groups not meeting their AMO. That is, proficient students in those 

groups who are projected to score below the cutoff in three years are not counted as proficient. 

A school in Tennessee making AYP is considered to have made it by growth if including 

 28 This approval stipulated that Tennessee work with the U.S. Department of Education to ensure that results from 

alternate assessments using alternate achievement standards were included by the 2006–07 school year.
Evaluation of the Growth Model Pilot Project 36

students projected to be proficient allows all non-proficient subgroups to meet the AMO. 

Tennessee includes subgroups in AYP determinations if they have at least 45 full academic year 

students and applies a 95 percent confidence interval to the AMO for status and safe-harbor 

determinations (like Arizona and Arkansas) instead of to the percent proficient.

Exhibit 19 shows that 1,130 (or 83 percent) of 1,357 Tennessee schools met AYP requirements 

in 2007–08 through status and safe-harbor criteria. An additional 22 schools (or 2 percent) that 

failed to meet these criteria made AYP by also applying the pilot growth model. A similar 

number of schools (19) made AYP by growth the previous year, though the total number of 

schools making AYP was higher (90 percent).

Exhibit 19

Tennessee School AYP Determinations With Status and Safe-Harbor

Results Augmented With Growth Model Results, 2006–07 and 2007–08

School AYP Under StatusPlus-Growth Model

Number Percent

2006–07 2007–08 2006–07 2007–08

Met with Status 1,149 964 84% 71%

Met with Safe-harbor 68 166 5% 12%

Met with Growth 19 22 1% 2%

Not Met 126 205 9% 15%

All Eligible Schools 1,362 1,357 100% 100%

Exhibit reads: For Tennessee’s schools overall, 964 met AYP under status in 2007–08, which 

was 71 percent of all eligible schools.

Source: U.S. Department of Education, EDFacts.

Impact of GMPP on AYP

As can be seen in Exhibits 10–19, the pilot states varied in the proportion of their schools making

AYP under status or safe-harbor; these differences can obscure the impact of their growth 

models on AYP. One way to assess the impact is to calculate the percentage increase in the 

number of schools making AYP due to use of the growth model. The schools making AYP 

uniquely by growth represented a percentage increase in the schools making AYP of 20 percent 

across all states, and ranged as high as 102 percent in Ohio, 24 percent in Florida, and 10 percent 

in Arkansas (Exhibit 20). Excluding Ohio, the overall percentage increase due to the GMPP in 

the other eight states was 5 percent.

Another indicator of the impact of the GMPP is the extent to which the growth models affected 

the pool of schools identified as failing both the status and safe-harbor criteria for AYP. Across 

all nine states, 16 percent of the 7,863 schools that did not make AYP by status or safe-harbor 

made AYP by growth (Exhibit 20). These rates ranged from a high of 50 percent of eligible 

schools in Ohio making AYP by growth to 5 percent or fewer of eligible schools in Alaska,

Arizona, and North Carolina making AYP by growth. Excluding Ohio, the overall percentage 

reduction due to the GMPP in the other eight states was just 4 percent compared to the 16 

percent when Ohio was included. As noted in the section above focusing on features of Ohio’s 

growth model, the main reason for the much higher impact of the growth model there on school 
Evaluation of the Growth Model Pilot Project 37

AYP was that the state used a much more inclusive definition of on-track to proficiency than the 

other pilot states. 

Exhibit 20

Percentage Increase in Number of Schools That Made AYP Due to Growth, and 

Percentage Decrease in Number of Schools That Did Not Make AYP Due to Growth, by 

State, 2007–08

Pilot States

Number of 

Schools 

Making AYP 

by Status or 

Safe-Harbor

Number of 

Schools not 

Making AYP by 

Status or SafeHarbor

Number of 

Schools Not 

Making AYP by 

Status or SafeHarbor That 

Met by Growth

Percentage 

Increase in 

Schools 

Making AYP 

Due to 

Growth

Percent of 

Schools Not 

Making AYP by 

Status or SafeHarbor That 

Met by Growth

All Nine States 6,213 7,863 1,246 20% 16%

Alaska 292 203 0 0% 0%

Arizona 1,117 379 8 1% 2%

Arkansas 500 390 52 10% 13%

Delaware 123 60 5 4% 8%

Florida 632 2,648 153 24% 6%

Iowa 721 377 23 3% 6%

North Carolina 737 1,612 0 0% 0%

Ohio 961 1,967 983 102% 50%

Tennessee 1,130 227 22 2% 10%

Exhibit reads: The 1,246 schools that made AYP by growth increased the number of schools making AYP from 

6,213 to 7,459 schools, which was a percentage increase of 20 percent. Of the 7,863 schools that did not make 

AYP under either status or safe-harbor, 1,246 or 16 percent made AYP using the growth model.

Source: U.S. Department of Education, EDFacts and the Delaware state department of education.

Impact of the GMPP on Subgroup AYP Outcomes

ESEA requires that each of a number of subgroups must reach AMOs in order for a school to 

receive AYP credit. This section documents the extent to which the growth models affect the 

AYP outcomes of each targeted subgroup. For each group, we present the numbers and 

percentages of subgroups that did not reach the AMO by the status model or safe-harbor that 

reached the AMO because of growth model. 

Although schools do not in most cases miss AYP simply because of the performance of a single 

subgroup,29

 29 See U.S. Department of Education, Office of Planning, Evaluation and Policy Development, Policy and Program 

Studies Service (2007). State and Local Implementation of the No Child Left Behind Act, Volume III—Accountability 

Under NCLB: Interim Report: 42, 43.

growth models may be especially beneficial to historically underperforming 

subgroups. This analysis provides a more fine-grained picture of the impact of the growth 

models in each state, pinpointing the rates at which each subgroup reached the AMO because of 

the growth model.
Evaluation of the Growth Model Pilot Project 38

The school-level AYP determinations and the methods of making AYP (status, safe-harbor, 

growth) examined thus far are based on results for the complete set of ESEA reporting subgroups 

represented in each school. The states also reported to EDFacts whether each subgroup was 

exempt by minimum group size criteria, met the AMO by status, met the AMO by safe-harbor, 

met the AMO by growth, or did not meet the AMO for both reading and mathematics. A 

subgroup met the AMO by growth if it did not make AYP by status or safe-harbor and sufficient 

numbers of below-proficient students were on-track to reach the AMO. 

The number and percentage of each subgroup that was classified as meeting the AMO in reading

by growth are shown in Exhibit 21 (results for mathematics are generally consistent with the 

reading results and are presented in Appendix C). In this analysis, schools are considered to be 

eligible to meet their AMO by growth if the respective reporting subgroup did not meet the 

AMO by either status or safe-harbor. The numbers of subgroups that met the AMO by growth 

are greater than the number of schools that made AYP by growth because schools may have one 

or more subgroups that met their AMO by growth but still had at least one other subgroup that 

did not meet the AMO by any method. If any subgroup within a school failed to meet the AMO

by any of the three methods, the school would not make AYP.

Exhibit 21

Number of Eligible Schools and Percentage in Which Reading AMO Was Met Because of 

Growth Model Results, by All Students and Racial or Ethnic Reporting Groups, 2007–08

Pilot States

Number of Eligible Schools*

All

Students White

Black, NonHispanic HispanicAsian/Pacific 

Islander

American Indian/

Alaskan Native

All Nine States 1,818 373 1,927 737 47 181

Alaska 74 3 3 0 8 68

Arizona 4 37 58 46 37 99

Arkansas 48 9 75 4 0 0

Delaware** 8 NA NA NA NA NA

Florida 849 108 922 484 1 0

Iowa 71 27 41 29 1 3

North Carolina 196 2 316 103 0 11

Ohio 529 171 487 69 0 0

Tennessee 39 16 25 2 0 0

Percent of Eligible Schools

All Nine States 18% 39% 15% 12% 0% 0%

Alaska 0% 0% 0% NA 0% 0%

Arizona 0% 0% 0% 0% 0% 0%

Arkansas 0% 0% 0% 0% NA NA

Delaware 0% NA NA NA NA NA

Florida 6% <1% 6% 9% 0% NA

Iowa 11% 11% 10% 7% 0% 0%

North Carolina <1% 0% <1% 0% NA 0%

Ohio 48% 74% 42% 61% NA NA

Tennessee 51% 88% 52% 0% NA NA

Exhibit reads: Across all nine states, among the 1,818 schools in which the ‘all students’ reporting group did not 

reach the reading AMO by either status or safe-harbor, that group did reach the AMO in 18 percent of the eligible 

schools when the growth criteria were applied.

* “Eligible schools” means schools in which the respective reporting group did not meet the reading AMO by 

status or safe-harbor and had grade levels to which the growth model was applied.

** EDFacts did not include subgroup information for Delaware in 2007–08.

NA: Not applicable due to no eligible schools. 

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 39

Exhibit 21 shows that subgroups meeting AMOs by growth were concentrated in Florida, Iowa, 

Ohio, and Tennessee, with the highest rates in Ohio and Tennessee. Overall, more black (1,927) 

and Hispanic subgroups (737) did not meet their reading AMO by status or safe-harbor than 

white subgroups (373), but lower percentages of the eligible black and Hispanic subgroups met 

the AMO when growth model results were factored in (15 percent of black and 12 percent of 

Hispanic subgroups versus 39 percent of the eligible white subgroups). Florida is an exception 

to this trend, with relatively fewer white subgroups (less than 1 percent) in eligible schools 

benefiting from the state’s pilot growth model than black (6 percent) and Hispanic subgroups (9

percent). Similarly for mathematics, relatively more eligible Hispanic (15 percent) subgroups 

met the AMO by growth than white (10 percent) subgroups in Florida (see Exhibit C.2).

Exhibit 22 extends the picture to the remaining three ESEA reporting subgroups, showing the 

impact of the GMPP on economically disadvantaged students, students with disabilities (SWD), 

and limited English proficient (LEP) students. In all nine states combined, 21 percent of lowincome subgroups that missed the reading AMO by status and safe-harbor were able to meet 

their AMO via the pilot growth models (results are similar for mathematics, see Exhibit C.3). 

Relatively fewer SWD (15 percent) and LEP student (6 percent) subgroups in eligible schools 

met the reading AMO by growth. The picture is slightly different in Florida and Iowa, where 

eligible LEP student subgroups met their AMO at higher rates from the GMPP than did lowincome student subgroups.

The higher rates of subgroups meeting both the reading and mathematics AMOs by growth in 

Ohio and Tennessee than in the other pilot states are noteworthy in that both states used a 

projection model methodology for making on-track to proficiency determinations. However, 

two factors make the impact of the projection model per se somewhat unclear. First, the relative 

size of the eligible pools in those two states were very different because of the difference in cut 

scores; Tennessee had relatively low cut scores and high rates of schools making AYP by status 

or safe-harbor, while Ohio had relatively high cut scores and much lower rates of AYP by status 

or safe-harbor. Second, Ohio adjusted its projection estimated scores upward by two standard 

error units, while Tennessee did not make such an adjustment. If Ohio had not made that 

adjustment, it is likely it would have had a much lower rate of on-track students among its nonproficient students and thus lower rates of subgroups meeting AMOs by growth and schools 

making AYP by growth.
Evaluation of the Growth Model Pilot Project 40

Exhibit 22

Number of Eligible Schools and Percentage in Which Reading AMO Was Met Because of 

Growth Model Results, by Low-SES, SWD, and LEP Reporting Groups, 2007–08

Pilot States

Number of Eligible Schools*

Economically 

Disadvantaged 

Students (low-SES)

Students with 

Disabilities (SWD)

Limited English 

Proficient (LEP) 

Students

All Nine States 2,562 3,017 833

Alaska 81 42 53

Arizona 71 282 157

Arkansas 76 70 6

Delaware** NA NA NA

Florida 1,075 1,185 463

Iowa 143 142 27

North Carolina 363 402 83

Ohio 719 881 43

Tennessee 34 13 1

Percent of Eligible Schools

All Nine States 21% 15% 6%

Alaska 0% 0% 0%

Arizona 0% 0% 0%

Arkansas 0% 0% 0%

Delaware NA NA NA

Florida 7% 4% 8%

Iowa 8% 15% 15%

North Carolina <1% 0% 0%

Ohio 58% 42% 19%

Tennessee 50% 15% 0%

Exhibit reads: Across all nine states, among the 2,562 schools in which the “economically 

disadvantaged” reporting group did not reach the reading AMO by status or safe-harbor, that 

group did reach the AMO in 21 percent of the eligible schools when the growth criteria were 

applied.

* “Eligible schools” means schools in which the respective reporting group did not meet the 

reading AMO by status or safe-harbor and had grade levels to which the growth model was 

applied.

** EDFacts did not include subgroup information for Delaware in 2007–08.

NA: Not applicable due to no eligible schools.

Source: U.S. Department of Education, EDFacts.

Discussion

The idea of using growth models to assess school academic performance is attractive and the 

data collection systems and database architecture needed to support such models are rapidly 

becoming available. The pilot models were strictly add-ons to the traditional status-plus-safeharbor models for determining AYP in all nine states. Even Delaware, the only state that applied 

the growth model and then augmented with the status model, used the growth model as an addon instead of a replacement. While states had the option of applying growth criteria instead of 

status and safe-harbor, none exercised that option. Growth criteria were instead added to status 

and safe-harbor criteria in ways that could only increase the number of schools making AYP.
Evaluation of the Growth Model Pilot Project 41

States also had the option of applying growth criteria before status and safe-harbor were applied, 

but Delaware was the only state that did so. Delaware’s growth-plus-status procedure had the 

effect of officially recognizing about 48 percent of their schools as making AYP by growth, 

which was a much higher percentage than under the status-plus-growth procedure used in the 

other states except Ohio. While this may have had a positive effect on the public visibility and 

understanding of the model in Delaware, the actual number of Delaware schools labeled as 

making AYP by growth that would not have made AYP under status or safe-harbor was small.

As shown in Exhibits 10 through 19, the percent of schools that made AYP using the growth 

provisions of the GMPP, and which would not have made AYP without those provisions, ranged

from a high of 34 percent of schools in Ohio to 0 percent in Alaska and North Carolina. Across 

all nine pilot states, most of the schools that made AYP by growth were located in Ohio; 

excluding Ohio, only 2 percent of all schools in the other eight states made AYP by growth.

Another perspective on these rates is gained when the number of schools making AYP under 

growth is compared not to the total number of schools in the state but instead to either the 

number of schools in the state that made AYP under either status or safe-harbor (the percentage 

increase due to growth) or the number that did not make AYP under either status or safe-harbor

(percentage decrease in non-AYP due to growth). The percentage increases in schools making 

AYP ranged from a high of 102 percent in Ohio to less than 2 percent in Alaska, Arizona, North 

Carolina, and Tennessee. The growth model decreased the number of schools identified as 

failing under the status-plus-safe-harbor model by 50 percent in Ohio but by less than 5 percent 

in Alaska, Florida, and North Carolina. The rate of making AYP by growth after missing by 

status and safe-harbor ranged from 6 percent to 13 percent in the remaining five pilot states.

Reasons for the variation among states in the percentages of schools making AYP by growth will 

be examined systematically in Chapter IV but are generally a combination of differences in the 

assessments used, the proficiency cutpoints, the features of the growth models implemented, the 

states’ AMOs, and the actual levels of growth realized by the students. It is noteworthy that the 

percentages making AYP by growth appear unrelated to the percentages making AYP by status 

or safe-harbor (both of which vary greatly among these states) or to whether the state adopted a 

transition matrix, trajectory, or projection type of growth model. The exceptionally high rate in 

Ohio is most likely explained by that state’s practice of augmenting students’ projected scores 

with statistically-derived quantities reflecting the uncertainty of the projected scores. In contrast, 

the very low rates in other states may reflect the impact of the various other (nongrowth) 

methods for determining AYP available in those states for schools to make AYP by status and 

safe-harbor (e.g., confidence intervals and multiyear averaging), such that the status and safeharbor methods picked up schools which would have made AYP by growth had those various 

provisions not been available. This issue is addressed in Chapter IV, where the numbers of 

schools in each state that could make AYP if growth model results were given priority over 

status and safe-harbor are estimated.
Evaluation of the Growth Model Pilot Project 42
Evaluation of the Growth Model Pilot Project 43

III. School Characteristics Associated with AYP Outcomes

This chapter addresses the question of whether certain school characteristics are associated with 

making AYP under growth but not status or safe-harbor. The school characteristics are 

organizational and demographic variables that include ESEA improvement status, poverty level, 

minority concentration, locality, and size. Of particular interest here is how pilot growth models 

affect the AYP designation of schools serving disadvantaged populations. Because such schools 

have been found to make AYP at much lower rates than those serving more affluent populations,30

The question addressed in this chapter is whether certain types of schools are more or less likely to 

make AYP by the growth criteria. To answer this question, the following analyses look at the 

percentage increase of schools of each type that made AYP because of the availability of the growth 

model under the GMPP. The percentage increase is calculated by dividing the number of schools 

making AYP by growth but not by status or safe-harbor by the number that made AYP by either 

status or safe-harbor. Any difference between comparison groups in the percentage increases of 

schools making AYP by growth will suggest that the GMPP had a disproportionate effect on certain 

types of schools. 

the GMPP may function to lessen the AYP gap on this dimension. 

The U.S. Department of Education report, State and Local Implementation of the No Child Left 

Behind Act, Volume II—Accountability Under NCLB: Interim Report (Le Floch, Martinez, O’Day, et 

al. 2007) found that certain school demographic characteristics were associated with the likelihood 

of making AYP. For example, the study found that high-poverty, high-minority, larger, and urban 

schools were less likely to make AYP by status criteria (Le Floch, Martinez, O’Day, et al. 2007: 39, 

40). Another important characteristic considered here is the ESEA improvement status 

classifications of schools. This chapter uses the data reported in EDFacts to characterize schools 

that made AYP under growth but not under status or safe-harbor. As discussed in Chapter II, 

Delaware reported results to EDFacts based on growth model results augmented with status and 

safe-harbor results. In order to identify the incremental impact of the growth model on schools’

likelihood of making AYP, we substitute data using status and safe-harbor augmented with growth 

for the Delaware EDFacts report. 

ESEA School Improvement Status.

 30 U.S. Department of Education, Office of Planning, Evaluation and Policy Development, Policy and Program Studies 

Service (2007). State and Local Implementation of the No Child Left Behind Act, Volume III—Accountability Under 

NCLB: Interim Report: 39.

An important issue for ESEA accountability is the extent to 

which schools identified for improvement are, despite their relatively low levels of performance, 

actually making progress toward the goal of universal proficiency. The growth model pilot is in part 

intended to identify such schools. States are required under ESEA to identify for improvement 

Title I schools that do not meet AYP for two consecutive years and to initiate a process of 

interventions designed to improve student outcomes; most states have applied a similar process for 

non–Title I schools. Schools not making AYP for two consecutive years are officially identified for 

improvement. If a school does not make AYP after two years in the improvement status, it is 
Evaluation of the Growth Model Pilot Project 44

identified for corrective action. If the school misses AYP for an additional year, it moves into 

restructuring status.31

As is shown in Exhibit 23 below, states vary with respect to whether the use of growth models 

allowed relatively more schools identified for improvement to make AYP than schools not identified 

for improvement. The percentage increase in making AYP due to growth is greater for identified 

versus unidentified schools in three states (Arizona, Iowa, and Ohio), lesser in two states (Arkansas 

and Delaware), and only slightly different in two states (Florida and Tennessee). In North Carolina, 

no schools made AYP by growth regardless of status relative to being identified for improvement. 

Improvement status data for Alaska was not included in EDFacts for year 2007–08.

Exhibit 23

Numbers of Schools Making AYP by Status or Safe-Harbor, and Percentage Increase in 

Schools Making AYP Due to Growth, by NCLB School Improvement Status, 2007–08

Pilot States

Numbers of Schools Making AYP 

by Status or Safe-harbor

Percentage Increase in Schools 

Making AYP Due to Growth

Identified for 

Improvement/ 

Under 

Corrective 

Action

Planning to 

Restructure 

or 

Restructuring

Not Identified 

for 

Improvement

Identified for 

Improvement/ 

Under 

Corrective 

Action

Planning to 

Restructure 

or 

Restructuring

Not Identified 

for 

Improvement

All Eight States 359 56 5,466 69% 48% 18%

Arizona 45 4 1,068 7% 25% <1%

Arkansas 76 10 414 5% 0% 12%

Delaware 3 0 120 0% 0% 4%

Florida 31 27 574 26% 48% 23%

Iowa 3 0 718 167% ∞* 2%

North Carolina 65 5 653 0% 0% 0%

Ohio 70 4 872 321% 300% 84%

Tennessee 66 6 1,047 3% 0% 2%

Exhibit reads: Of the schools across all eight pilot states that are either identified for improvement or under corrective 

action in 2007–08, 359 make AYP by status or safe-harbor and the number making AYP is increased 69 percent by 

schools making AYP by growth.

*

Source: U.S. Department of Education, EDFacts.

One school in Iowa that was planning to restructure or in the process of restructuring made AYP by growth but none 

made AYP by status or safe-harbor. The percentage increase due to growth could not be calculated because the 

denominator was zero.

School Poverty Concentration. The 2007 Interim Report on the implementation of NCLB found 

that schools’ likelihood of making AYP under the status model was strongly correlated with student 

socioeconomic status and racial or ethnic minority representation.32

 31 U.S. Department of Education, Office of Planning, Evaluation and Policy Development, Policy and Program Studies 

Service (2007). State and Local Implementation of the No Child Left Behind Act, Volume III—Accountability Under 

NCLB: Interim Report: 3-6.

 Exhibit 24 below shows that the 

32 Ibid, 39. 
Evaluation of the Growth Model Pilot Project 45

GMPP reduced the gap between low- and high-poverty schools.

33

Exhibit 24

Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and Percentage Increase 

in AYP Due to Growth, by School Poverty Concentration, 2007–08

 Expressed as a percentage 

increase, the pilot growth models increased the number of high-poverty schools making AYP by 20 

percent while increasing the number of low-poverty schools making AYP by 18 percent. There is 

considerable variation by state, however. The effect of growth on high- compared with low-poverty 

schools making AYP was much higher in Florida (71 percent versus 10 percent), Iowa (20 percent 

versus 3 percent), Ohio (169 percent versus 68 percent) and Tennessee (6 percent versus <1 percent). 

Growth models had little or no differential impact on high-poverty schools making AYP in Alaska, 

Arizona, Arkansas, Delaware, and North Carolina.

Pilot States

Numbers of Schools Making AYP 

by Status or Safe-Harbor

Percentage Increase in Schools 

Making AYP Due to Growth

Low Poverty Medium Poverty High Poverty Low Poverty Medium Poverty High Poverty

All Nine States 1,946 3,402 826 18% 21% 20%

Alaska 276 0 16 0% 0% 0%

Arizona 315 520 281 <1% <1% 1%

Arkansas 25 389 86 8% 11% 8%

Delaware 29 90 4 0% 6% 0%

Florida 272 315 45 10% 30% 71%

Iowa 265 451 5 3% 3% 20%

North Carolina 187 471 61 0% 0% 0%

Ohio 458 439 64 68% 128% 169%

Tennessee 119 727 264 <1% <1% 6%

Exhibit reads: Of the schools across all nine pilot states that had low enrollments of children from poverty-level 

households in 2007–08, 1,946 make AYP by status or safe-harbor and the number making AYP is increased 18 percent

by schools making AYP by growth.

Source: U.S. Department of Education, EDFacts.

School Minority Concentration. Results for school minority composition are shown in Exhibit 

25.

34

 33 Poverty level is defined here in terms of the percentages of students eligible for the federal free and reduced-price 

lunch program (FRPL). Low-poverty schools enroll 25 percent or fewer FRPL-eligible students, medium-poverty 

schools enroll 26 percent to 75 percent FRPL-eligible students, and high-poverty schools enroll more than 75 percent.

Across all nine states the use of pilot growth models appears to benefit low-minority schools 

more than high-minority schools (23 percent versus 18 percent). Analyzed separately, however, all 

states show equal or greater growth impact for high-minority schools (an example of the 

“aggregation paradox”). The gap in making AYP between high- and low-minority schools was 

reduced in Arizona, Arkansas, Florida, Ohio, and Tennessee. In these states the percent increase for

high-poverty schools making AYP by growth was considerably higher than the percent increase for

low-minority schools. Two states—Delaware and Iowa—did not have any high-minority schools 

34 Minority level is defined here in terms of the percentages of non-white or Hispanic students. Low-minority schools 

enroll 25 percent or fewer non-white or Hispanic students, medium-minority schools enroll 26 percent to 75 percent nonwhite or Hispanic students, and high-minority schools enroll more than 75 percent.
Evaluation of the Growth Model Pilot Project 46

but displayed a similar pattern in having the use of growth models close the AYP gap among 

medium- and low-minority schools. In Alaska and North Carolina, no schools made AYP by growth 

regardless of minority composition. 

Exhibit 25

Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and Percentage Increase 

in AYP Due to Growth, by School Minority Concentration, 2007–08

Pilot States

Numbers of Schools Making AYP by Status or 

Safe-Harbor

Percentage Increase in Schools Making AYP 

Due to Growth

Low Minority Medium Minority High Minority Low Minority Medium Minority High Minority

All Nine States 3,894 1,553 645 23% 14% 18%

Alaska 123 78 86 0% 0% 0%

Arizona 368 499 213 <1% <1% 2%

Arkansas 343 126 22 8% 15% 14%

Delaware 31 86 6 0% 6% 0%

Florida 275 236 110 11% 33% 37%

Iowa 684 21 0 3% 14% 0%

North Carolina 449 229 43 0% 0% 0%

Ohio 854 59 29 95% 180% 197%

Tennessee 767 219 136 <1% 4% 6%

Exhibit reads: Of the schools across all nine pilot states that had low enrollments of minority children in 2007–08, 3,894 

make AYP by status or safe-harbor and the number of schools making AYP is increased 23 percent by schools making 

AYP by growth.

Source: U.S. Department of Education, EDFacts.

School Urbanicity. School location in an urban, suburban, or rural area is another variable that 

previous studies have found related to AYP results, with rural and suburban schools generally more 

likely to make AYP under status or safe-harbor than urban schools. Exhibit 26 shows that, in most 

states, the growth model moved relatively more urban schools than suburban or rural schools into 

making AYP. Iowa was the only state among the nine to exhibit the opposite trend, while the GMPP 

tended to benefit suburban (28 percent) versus urban (23 percent) schools in Florida. Again, Alaska 

and North Carolina had no schools making AYP by growth in any category. 
Evaluation of the Growth Model Pilot Project 47

Exhibit 26

Numbers of Schools Making AYP Under Status-Plus-Safe-Harbor, and Percentage Increase 

in AYP Due to Growth, by School Urbanicity, 2007–08

Pilot States

Numbers of Schools Making AYP

by Status or Safe-Harbor

Percentage Increase in Schools 

Making AYP Due to Growth

Urban 

Schools

Suburban 

Schools

Rural

Schools

Urban 

Schools

Suburban 

Schools

Rural 

Schools

All Nine States 1,327 1,341 3,460 16% 37% 15%

Alaska 34 6 248 0% 0% 0%

Arizona 471 213 399 1% <1% <1%

Arkansas 73 35 383 19% 14% 8%

Delaware 13 57 53 23% 2% 2%

Florida 152 327 146 23% 28% 16%

Iowa 75 43 597 1% 5% 3%

North Carolina 119 112 503 0% 0% 0%

Ohio 105 378 464 135% 103% 95%

Tennessee 285 170 667 3% 1% 2%

Exhibit reads: Of the schools across all nine pilot states that were located in urban areas in 2007–08, 1,327 make AYP 

under status or safe-harbor and the number of schools making AYP is increased 16 percent by schools making AYP by 

growth.

Source: U.S. Department of Education, EDFacts.

Discussion

The answer to the question, “Is the likelihood of making AYP under growth models associated with 

school characteristics?” varied across the pilot states and also varied within states across years. 

Exhibit 27 summarizes the results in Exhibits 23 through 26 above by showing whether the GMPP 

reduced the AYP gap (√) for that type of school compared with schools at the other end of the 

spectrum (e.g., high- vs. low-poverty schools). Results for 2006–07 published in the Interim Report 

on the Evaluation of the Growth Model Pilot Project (2010) are also added to compare the impact of 

the pilot program across years. The results summarized here are all bivariate relationships and do 

not indicate independent effects of the school characteristics based on statistical adjustments for the 

substantial overlaps among the characteristics (for example, the tendency of schools identified for 

improvement to also enroll high proportions of low-income minority students). 
Evaluation of the Growth Model Pilot Project 48

Exhibit 27

Summary of the Effect of GMPP by School Demographic Characteristic and by State, 

2006–07 and 2007–08

Pilot States

Identified for 

Improvement / 

Under Corrective 

Action

High Poverty 

School

High Minority 

School

Urban School

(compared to 

suburban 

schools)

2006–07 2007–08 2006–07 2007–08 2006–07 2007–08 2006–07 2007–08

All States √ √ √ √ √ - - -

Alaska n/a n/a - - - - - -

Arizona - √ - √ - √ - √

Arkansas √ - √ - √ √ √ √

Delaware - - - - - - - √

Florida √ √ √ √ √ √ √ -

Iowa √ √ √ √ √ - √ -

North Carolina √ - √ - √ - √ -

Ohio n/a √ n/a √ n/a √ n/a √

Tennessee √ √ √ √ √ √ √ √

Exhibit reads: For all states in the 2007–08 school year, pilot growth models reduced the AYP gap between 

schools identified for improvement or under corrective action and schools not so identified; reduced the AYP 

gap between high-poverty and low-poverty schools; did not reduce the AYP gap between high-minority and 

low-minority schools; and did not reduce the AYP gap between urban and suburban schools. Note: “ √” 

indicates the AYP gap was reduced, while “-” indicates that the AYP gap was not reduced.

Source: U.S. Department of Education, EDFacts and the Delaware state department of education.

Looking first at schools identified for improvement under ESEA for 2007–08, the percentage 

increase in such schools making AYP due to the growth models was greater than among schools

not identified for improvement in Arizona, Iowa, and Ohio. The GMPP favored schools not 

identified for improvement in Arkansas and Delaware but did not favor either type of school in 

Florida and Tennessee. Overall, schools identified for improvement benefitted from the pilot 

program relative to schools not identified in both 2006–07 and 2007–08, though the effect 

differed between years for Arizona, Arkansas, and North Carolina. 

Turning to school demographic characteristics, the effect of growth models on AYP was greater 

among schools enrolling higher proportions of students from poverty-level households in both 

2006–07 and 2007–08. The pilot program was particularly effective in reducing the gap between

the percent of low- and high-poverty schools making AYP in Florida, Iowa, Ohio, and 

Tennessee. Once again the effect differed between years for Arizona, Arkansas, and North 

Carolina. The growth component also was more likely to identify schools as making AYP 

among those with higher concentrations of minority students compared to schools with lower 

concentrations in the majority of GMPP states, though the effect differed by school year across 

all pilot states and for Iowa and North Carolina. The growth component did not benefit urban 

relative to suburban schools in either 2006–07 or 2007–08 but did close the gap in the majority 

of states in both years, though the effect differed across years for the majority of pilot states. 
Evaluation of the Growth Model Pilot Project 49

IV. Characteristics of Growth Models Affecting AYP Outcomes

The results presented in Chapters II and III show several large differences among the pilot states 

in the proportions of schools making AYP by growth as well as by status and safe-harbor. These 

differences are likely to arise from a number of sources, including features of the states’

assessment systems, methods of determining AYP under status and safe-harbor, and 

characteristics of the growth models themselves. This chapter focuses on technical features of 

the growth models and their effect on AYP outcomes across the nine pilot states.

The school-level AYP determinations analyzed in Chapters II and III are aggregates of AYP 

determinations for subgroups, and AYP determinations for subgroups are aggregates of status 

and growth determinations for individual students. The growth model results for individual 

students were not collected for the EDFacts repository. However, the GMPP states were 

required to compile growth model results for all students in the participating grade levels. 

Specifically, each of the pilot states included in this Final Report provided student-level data 

with variables indicating (1) whether the student scored at the proficient level or higher on the 

state’s reading and mathematics tests, and (2) whether the student was on-track to attain (or 

maintain) proficiency in the two subjects according to the growth model. In addition, some 

states were also able to provide scale score results for each student. 

The analyses presented in this chapter draw on these student-level data to address several

questions related to how and whether AYP determinations would be affected if the data collected 

by the GMPP states were used in various different ways. Six specific questions are addressed:

1. How do the nine state models compare in terms of the number of students they 

classify as on-track to proficiency? 

2. How would the number of schools identified as making AYP by growth change in the 

nine pilot states if the growth model results were applied before status and safeharbor criteria were used to assess AYP?

3. How do the three main types of growth models (transition matrix, trajectory, and 

projection) compare in terms of the number of students they identify as on-track and 

the number of schools that meet their AMO by growth criteria? 

4. How do the main types of growth models compare in terms of their predictive 

accuracy of on-track students actually attaining proficiency? 

5. How would AYP results change if the growth standards were not tied to students 

attaining proficiency standards? 

6. To what extent did the pilot states have test scores for two or more years for all 

students, and did the non-matched students differ from their matched counterparts? 

Comparison of Student On-Track and Proficiency Results

As discussed in Chapter I with respect to schools, it is theoretically possible for a student to have 

different proficiency determinations for growth and status. These possibilities are illustrated in 
Evaluation of the Growth Model Pilot Project 50

Exhibit 28, which outlines the four possible patterns. A student can be proficient under status

and on-track to proficiency per the growth model (cell “A”), on-track to proficiency under the 

growth model but not proficient (cell “B”), proficient under status but not on-track to maintain 

proficiency (cell “C”), or neither proficient under status nor on-track to proficiency (cell “D”). 

The primary goal of the GMPP growth models is to identify students in cell B and to allow the 

schools to count those students the same as proficient students for accountability purposes.

Although growth models allow for the identification of students in cell C as well, these students 

did not contribute negatively to AYP decisions for any school under the GMPP.

Exhibit 28

Conceptual Map of How Growth Model On-Track to Proficiency Designations Compare 

With Status Model Proficiency Designations for Students

Student On-Track to Proficiency 

Under the Growth-Only Model

Student Proficiency Under the Status Model

Proficient Not Proficient

On-Track to Reach or Maintain Proficiency A B

Not On-Track to Reach or Maintain Proficiency C D

The percentages of students in each cell are shown in Exhibit 29. The percentages in cell A 

(proficient or higher and on-track to maintain or exceed proficiency) range from lows of 44 

percent in Florida, 48 percent in North Carolina, and 49 percent in Arkansas to about two-thirds 

in Alaska, Delaware, Iowa, and Ohio and as high as 89 percent in Tennessee. The percentages of 

students classified as proficient but not on-track to maintain that level (cell C) divide into five 

states with 1 percent or fewer and four states with 7–9 percent. The groups largely coincide with 

the types of growth models (described in Chapter I) implemented: the states using transition

matrix (Delaware and Iowa) and projection models (Ohio and Tennessee) have 0 percent in cell 

C while all of the states using trajectory models except Alaska have non-zero percentages. 

Alaska and the two transition matrix states opted to categorically define all proficient students as 

also on-track to proficiency. The projection model states did not impose such a rule but, as will 

be developed in the section of this chapter comparing generic versions of the models, their 

methods of determining whether students were on-track have a predictable tendency to classify 

proficient students as on-track. This tendency was amplified in Ohio by its procedure of 

adjusting upward the projected scores by two standard error units.

The main expectation is that the growth model will identify some students in cell B (not 

proficient but on-track) and thus increase the percentage of students that contribute to positive 

AYP determinations (the sum of cells A and B). Comparing the percentages of students in cell 

B, all states except Ohio had less than 10 percent of their students in this category. The 

variations among states in the cell B percentages are likely to reflect a combination of factors, 

including the effectiveness of the states’ systems in moving below-proficient students to the ontrack level as well as differences in the pool of eligible (non-proficient) students and the 

assessment instruments and relative difficulty of meeting the proficiency standards. 

The cell B results can be standardized to some extent by viewing them in relation to the total 

pool of non-proficient students (the sum of cells B and D). Dividing the cell B percentage by the 
Evaluation of the Growth Model Pilot Project 51

sum of cells B and D gives the rate at which non-proficient students are classified as on-track. 

This rate was by far the highest in Ohio where 75 percent of the non-proficient students were 

classified as on-track. The next highest rate was 45 percent in Tennessee, the other state using a 

projection model. Beyond those two states, Arizona had the next highest rate at 25 percent 

followed by Delaware and North Carolina at 18 percent, Florida at 13 percent, Arkansas and 

Iowa at 9 percent, and Alaska at 6 percent.

While Ohio had both the highest percentage in cell B and the highest rate of schools making 

AYP by growth documented in Chapter II, the relative sizes of the cell B percentages in the other 

states do not exactly correspond to their percentages of schools that made AYP by growth. 

Arizona and North Carolina, for example, had less than 1 percent of their schools classified as 

making AYP by growth despite having relatively high rates of non-proficient students identified 

as on-track to proficiency. The lack of correlation between the student and school results 

generally means the states differed in how the cell B students figured into the AYP 

determinations. For example, the pilot states differed in how they used confidence intervals and 

safe-harbor provisions, such that the impact of adding on-track students on AYP outcomes varied 

from state to state. 

Exhibit 29

Distribution of Students According to How Their Proficiency and On-Track 

to Proficiency Classifications Compare, 2007–08

Proficient and 

On-Track

(Cell A in 

Exhibit 25)

Not Proficient 

but On-Track 

(Cell B in 

Exhibit 25)

Proficient but 

Not On-Track 

(Cell C in 

Exhibit 25)

Neither 

Proficient nor 

On-Track

(Cell D in 

Exhibit 25)

Number of 

Students

Alaska 68% 2% 0%* 31% 68,787

Arizona 56% 9% 9% 27% 288,380

Arkansas 49% 4% 8% 39% 161,962

Delaware 67% 6% 0%* 27% 71,641

Florida 44% 6% 8% 42% 1,446,886

Iowa 68% 3% 0%* 29% 146,983

North Carolina 48% 8% 7% 37% 412,072

Ohio 68% 24% 0% 8% 584,988

Tennessee 89% 5% 0% 6% 401,659

Exhibit reads: Of the 68,787 students in the state of Alaska, 68 percent were proficient and on track to 

remain at or above the proficiency cut point, 2 percent were not proficient but were on-track to reach 

proficiency, 0 percent were proficient but not on-track to remain at or above the proficiency cut point, and 

31 percent were neither proficient nor on track to proficiency.

* In the models employed by Alaska, Delaware, and Iowa, all proficient students are defined as being ontrack.

Source: U.S. Department of Education, EDFacts and the Alaska, Arizona, Arkansas, Delaware, Florida, Iowa, North 

Carolina, Ohio, and Tennessee state departments of education.
Evaluation of the Growth Model Pilot Project 52

Effects of Order of Application of Growth, Status, and Safe-Harbor

As discussed in Chapter I, the pilot states had the option to apply growth criteria to school AYP 

determinations in a number of ways: after status and safe-harbor, before status and safe-harbor, 

without status or safe-harbor, etc. Of the nine pilot states in 2007–08, all but Delaware elected to 

apply growth criteria after status and safe-harbor. Delaware, in contrast, applied growth criteria 

first, followed by status and then safe-harbor. If all three criteria (status, safe-harbor, and 

growth) are applied, the total number of schools making AYP is not affected by changes in the 

order in which the criteria are applied. Nonetheless, the quality of the information about the 

schools provided by the AYP data is arguably affected by the order of application, depending on 

the manner in which the AYP data are reported. This analysis assesses the extent to which 

changes in order of application would lead to states having higher rates of schools classified as 

making AYP by their growth models.

Theoretically, schools that would make AYP by growth-only would be more likely to attain or 

maintain AYP in the future than schools that make AYP by status but not growth, and also more 

likely to attain or maintain AYP in the future than schools that make AYP by safe-harbor but not 

by status or growth. If the AYP results are reported in a way that shows the criteria under which 

a school first achieves the AMO benchmark, applying growth criteria before status and safeharbor could thus provide policymakers and the public with better information about school 

performance even though the overall numbers of schools making AYP would be the same. This 

is significant in light of policymakers’ interests in possibly making greater use of student growth 

data, for it will identify schools that would have made AYP if growth were used not just before 

but also instead of status or safe-harbor. 

The results in Chapter II indicated that, in most of the pilot states, relatively few additional 

schools made AYP by growth after status and safe-harbor criteria were applied. The logical 

possibilities of how hypothetical AMO results based solely on the on-track to proficiency data 

can intersect with actual AYP designations are shown in Exhibit 30. Referring to the cells in this 

table, the analysis in Chapter II focused on the number of schools that made AYP by growth 

(cells C plus G). However, those results do not provide a comprehensive account of the growth 

model data collected through the GMPP. First, it was possible for the schools that made AYP by 

status or by safe-harbor to also have had sufficient numbers of students classified by the growth 

model as on-track to proficiency to also meet their AMOs strictly under the growth criteria. 

Those schools would be located in cells A and B in Exhibit 30. Second, it was also possible for 

the schools that were classified as having made AYP by status-plus-growth to either meet (cell 

C) or not meet (cell G) the AMO if the students’ on-track to proficiency data were used 

exclusively. Not meeting the AMO in this case would mean that a sufficient number of students 

testing as proficient (or above) are classified as non-proficient because they are not on-track to 

maintain proficiency.
Evaluation of the Growth Model Pilot Project 53

Exhibit 30

How the Growth Model On-Track-to-Proficiency Designations Can Compare With AYP 

Designations for Schools

School AMO Designations Under a 

Hypothetical Growth-Only Model

School AYP Designations in ESEA Reporting, Based on 

Status-Plus-Growth Determinations

Made AYP by 

Status

Made AYP by 

Safe-Harbor

Made AYP by 

Growth

Did not Make 

AYP

Met AMO with growth-only A B C D

Did not meet AMO with growth-only E F G H

As noted in Chapter I, the EDFacts data classified schools in terms of AYP (and reporting 

groups in terms of AMO) with sets of mutually exclusive categories that do not allow one to 

address hypothetical questions of what would happen if the student growth model results were 

used differently. In order to assess the possible magnitude of the cells in Exhibit 27, the best 

resource is the status and growth proficiency data provided in the student files compiled by each 

of the pilot states. These data do not provide sufficient information to implement the states’

methodologies for determining whether schools made AYP, but they do contain information on 

the key dimensions of reading and mathematics proficiency and thus allow assessments of 

whether reporting groups and schools as wholes met their AMOs. 

In Alaska, Delaware, and Iowa, all students who scored at proficiency or higher on the status 

criteria were automatically also classified as “on-track” to proficiency per their growth models. 

The other six states, in contrast, applied their growth models to all students and identified at least 

some proficient students who were not on-track to continue to score at or above the proficiency 

levels in later grades.

35 It is important to note that in Alaska, Delaware, and Iowa it is possible 

for schools that made AYP by status to not meet the AMO using the growth model on-track 

indicator, even though all students who are proficient or higher were automatically counted as 

on-track as well. This is because these states identified a number of schools as making AYP by 

status that did not meet their AMO but that were close enough to count as making AYP because 

of the use of confidence intervals and multiyear averaging.

Status AYP and Growth-only Results

 35 There were a few exceptions to this rule in that some states did not actually calculate on-track indicators for 

students in certain grades. In those grades, the analyses reported in Exhibits 31 and 32 used the proficiency 

indicators for students in lieu of the on-track data in the calculation of the “growth-only” rates. This was done 

(instead of excluding those students) in order to improve comparability with the EDFacts data used in these 

analyses. Including versus excluding was particularly consequential in Ohio, which did not apply the growth model 

to third-graders and which had over half of the schools that failed to make AYP per EDFacts do so because third 

graders did not meet their AMOs in reading or mathematics.

. The results shown in the first column of Exhibit 31

indicate that for all nine states combined, 62 percent of the schools that made AYP by status also 

met their reading and mathematics AMOs using just the growth criteria. The states varied 

considerably, ranging from only 46 percent in Arizona and 47 percent in Arkansas and North 

Carolina to 75 percent or more in Ohio, Delaware, and Tennessee.
Evaluation of the Growth Model Pilot Project 54

Exhibit 31

Percentage of Schools Meeting AMO Using Only the Growth Model On-Track Indicator, 

by Standard EDFacts AYP Classification and State, 2007–08

Pilot States

Making AYP

by Status

Making AYP

by Safe-Harbor

Making AYP

by Growth

All Schools 

Making AYP

Not Making 

AYP

All Nine States 62% 28% 31% 51% 3%

Alaska 54% 4% ND 42% * 7%

Arizona 46% 17% 13% 43% 5%

Arkansas 47% 64% 23% 56% 3%

Delaware 79% ND 20% * 77% 24%

Florida 58% 16% 20% 45% 2%

Iowa 57% 30% 35% 54% 15%

North Carolina 47% 3% ND 23% * <1%

Ohio 75% 45% 34% 52% <1%

Tennessee 81% 5% 32% 69% 7%

Exhibit reads: Across all nine states, 62 percent of the schools that were classified in EDFacts as making 

AYP by status also met their AMOs when only the growth model on-track to proficiency indicator was used. 

Twenty-eight percent of the schools that were classified in EDFacts as making AYP by safe-harbor also met 

their AMOs when only the growth model on-track-to-proficiency indicator was used, and 31 percent of the 

schools listed as making AYP by growth met their AMOs using strictly the growth indicator.

*

Source: U.S. Department of Education, EDFacts and the Alaska, Arizona, Arkansas, Delaware, Florida, Iowa, North 

Carolina, Ohio, and Tennessee state departments of education.

“ND” stands for “not defined;” there were no schools in these cells.

Taken at face value, these results indicate that the great majority of schools that made AYP by 

status in Ohio, Delaware, and Tennessee would have also met their AMO using only the on-track 

to proficiency data, but only 46 to 58 percent of the schools in the other seven pilot states that 

made AYP by status would have met their AMO using growth-only. 

These findings provide some evidence that despite the low rates of making AYP by growth 

found in most states and documented in Chapter II, many schools had sufficiently high rates of 

students being on-track to proficiency to meet their AMOs using growth only. Under the normal 

practice of applying status before growth, the large numbers of schools that could have met their 

AMOs by growth-only were obscured.

Safe-harbor AYP and Growth-only Results. The second column of numbers in Exhibit 31

shows the percentages of schools that, according to EDFacts, made AYP by safe-harbor which 

also would meet the AMO for reading and mathematics using only the student on-track to 

proficiency indicator. The safe-harbor provision of ESEA was designed to identify schools 

making progress toward meeting the AMO even though their reading or mathematics proficiency 

levels, or both, are still below the AMO. Specifically, safe-harbor recognizes schools and 

subgroups that have decreased the percentage of students scoring below proficiency thresholds 

by 10 percent or more from one year to the next (e.g., decreasing the percentage of nonproficient students from 30 percent to 27 percent) and also make progress on the “other academic 

indicator,” typically the average daily attendance rate for elementary and middle schools and the 

graduation rate for high schools. Identifying progress among low-performing schools is, of 
Evaluation of the Growth Model Pilot Project 55

course, also the intention of the growth models, but the growth models are distinguished by (1) 

basing progress estimates on longitudinal student-level data, and (2) evaluating progress in terms 

of whether it is sufficient to reach the AMO within a specific time frame (e.g., by the time the 

student completes grade 8). By virtue of both these distinguishing features, schools meeting 

their AMOs by growth should be performing better than schools meeting their AMOs by safeharbor but not by growth. Nonetheless, all of the pilot states except Delaware applied safeharbor provisions before growth criteria in their AYP determinations. 

The results in Exhibit 31 indicate that only 28 percent of the safe-harbor schools in the eight

states that had any safe-harbor schools also met their AMO using the growth criteria. The 

percentages did not exceed 30 percent in any states except Arkansas (64 percent) and Ohio (45 

percent).

The relatively low level of overlap of safe-harbor and growth-only outcomes in most states 

suggests that safe-harbor often identified schools as improving when the growth model indicated 

otherwise. The growth-only percent of students on-track could be a better gauge of actual test 

score improvement than safe-harbor because the growth-only measure relies on the longitudinal 

student records rather than the percentages proficient in each year. This is not to say that the 

safe-harbor schools that would not have met AMOs using growth-only were not making 

improvements but only to indicate instead that they were not making as much improvement as 

the growth-only measure requires. Insofar as the growth criteria are more informative about 

school improvement, they could be usefully applied to AYP determinations before safe-harbor is 

applied.

The third column in Exhibit 31 shows the percentages of schools classified as making AYP by 

growth in EDFacts that also met their reading and mathematics AMOs using only the student ontrack to proficiency indicator. It is possible for these “made by growth” schools to not reach 

their AMOs using only growth criteria because the EDFacts classifications were based on statusplus-growth criteria instead of growth-only criteria. More specifically, the growth-only criteria 

classify some students who scored at or above the proficiency cut score as not on-track to 

proficiency and this makes the percentage of on-track lower than the percentage of proficientplus-on-track. The results indicate that only 31 percent of the “made by growth” schools would 

have met their AMOs if only the on-track to proficiency indicator were used. 

The fifth column in Exhibit 31 illustrates the lack of agreement between school-level results 

reported in EDFacts and those obtained from aggregating the student data within this evaluation 

project. Theoretically, the only schools that should show up in this column would be ones that 

met their AMOs for all subgroups but who failed to make AYP because of other criteria 

including participation rates and “other academic indicator” (average daily attendance for 

elementary and middle schools) results. This expectation appears to be born out in the overall 

rate of just 3 percent and the low rates in most states. However, the high rates for Delaware (24

percent) and Iowa (15 percent) suggest that other factors are complicating matters for those 

states, either on the side of how they made school AYP determinations for EDFacts or on the 

side of how they calculated the student on-track indicators provided to the evaluation team, or 

both.
Evaluation of the Growth Model Pilot Project 56

Exhibit 32 shows the results of applying the growth-only criterion before status and safe-harbor

within a framework of mutually exclusive categories like the ones currently used in EDFacts. 

For these calculations, the schools identified in EDFacts as not making AYP under any method 

were automatically kept as “not making AYP” in Exhibit 32 and the growth-only results applied 

to the schools classified in EDFacts as making AYP by status, safe-harbor, or status-plusgrowth. These figures show that 27 percent of all schools in the pilot states would meet AMOs 

using growth-only, while the percent making it strictly by status would fall to 13 percent. While 

the overall rate of schools not making AYP (46 percent) would be about the same as under the 

status-plus-growth framework, the use of growth criteria before the others gives a clearer picture 

of the extent to which schools are attaining growth-to-proficiency goals than the status-plusgrowth method.

Exhibit 32

Percentage of Schools Meeting the AMO Using Growth-Only and Making AYP by Other 

Means After Growth-Only Is Applied, by State, 2007–08

Pilot States

Meeting AMO 

by GrowthOnly

Making AYP 

by Status

Making AYP 

by SafeHarbor

Making AYP 

by Status 

plus Growth

Not 

Making 

AYP Total

Number of 

Schools

All Nine States 27% 13% 7% 6% 46% 100% 13,401

Alaska 25% 21% 14% 0% 41% 100% 490

Arizona 32% 37% 5% <1% 25% 100% 1,406

Arkansas 35% 8% 15% 4% 38% 100% 890

Delaware 54% 14% 0% 2% 30% 100% 182

Florida 11% 7% 3% 4% 76% 100% 3,280

Iowa 36% 26% 3% 1% 33% 100% 1,059

North Carolina 7% 7% 17% 0% 69% 100% 1,842

Ohio 34% 7% 3% 22% 33% 100% 2,896

Tennessee 58% 14% 12% 1% 15% 100% 1,356

Exhibit reads: In all nine pilot states, 27 percent of the schools would have met their AMO using only the growth 

models’ on-track to proficiency results. Thirteen percent of the schools would have made AYP by status after applying 

the growth-only determination, 7 percent would have made AYP only by safe-harbor, 6 percent would have made AYP 

by status-plus-growth, and 46 percent would not have made AYP.

Source: U.S. Department of Education, EDFacts and the Alaska, Arizona, Arkansas, Delaware, Florida, Iowa, North Carolina, Ohio, 

and Tennessee state departments of education.

Effects of Types of Models

As described in Chapter I, the GMPP states used a variety of growth models to determine 

whether students were on-track to maintain or attain proficiency. These models were classified 

into three general types: transition matrix models, trajectory models, and projection models. By 

requirement of the pilot project, these models had to conform to seven core principles, including 

universal proficiency by 2014 (see Exhibit 3). This section contrasts these three types of growth 

models by using simplified model formulations and applying them to a common data source. 

This approach controls many of the nongrowth factors that can confound cross-state differences 

and allows generalizable conclusions about growth model features. No judgment is made about 

which models are preferable, and, certainly, the universe of possible models is larger than these 
Evaluation of the Growth Model Pilot Project 57

three types. Instead, this section describes contrasting model features as they may support the 

contrasting policy priorities of individual states. 

One of the findings from the preceding chapters of this report is that a number of nongrowthrelated policy factors can moderate the impact of growth models. These factors are numerous 

and include safe-harbor, confidence intervals, AMOs, and cut scores. Unconditional cross-state 

differences between growth model results may be as much a function of these factors as of the 

types of models. To isolate the effects of the models, this section describes “generic” versions of 

the transition matrix, trajectory, and projection models used in the pilot states and applies them 

to longitudinal, standardized data from a single state: North Carolina.

The section begins with a description of the dataset and the parameters of three generic models. 

These generic models were chosen to be both representative of the state models in practice yet 

simple enough to be generalizable. The section continues with a comparison of student-level 

classifications under these three generic models. This comparison is performed for both a statusplus-growth approach, in which only non-proficient students are growth-eligible, and a growthonly model, in which proficient and non-proficient students must both make adequate growth. 

The section provides explanations and interpretations of differences among the generic models in 

the numbers and types of students they identify as on-track. The final part of the analysis 

aggregates the student-level results from each of the generic growth models to contrast school 

AYP results under the three models.

The questions addressed in this section and brief summaries of the findings with respect to 

growth model types are outlined in Exhibit 33. 

Exhibit 33

Summary of Research Questions About the Generic Types of Growth Models

Research Questions Findings

Which model is most likely to identify non-proficient

students as on-track?

The trajectory and transition matrix models are more likely than 

the projection model. The contrast increases with higher (more 

stringent) proficiency cut scores.

Which model is most likely to identify proficient 

students as on-track?

The projection model is more likely than the trajectory and 

transition matrix models. 

Which model is most likely to reclassify a school 

from not making AYP under status to making AYP 

when growth results are added to status results?

The trajectory model is slightly more likely to reclassify schools 

from non-AYP to AYP under a status-plus-growth method. This 

contrast increases with higher (more stringent) proficiency cut 

scores.

Which model is most likely to classify a school as 

making AYP using a growth-only criterion?

The projection model is much more likely to classify schools as 

making AYP under a growth-only criterion. This contrast 

decreases with higher (more stringent) proficiency cut scores.

Under which model are students identified as ontrack to proficiency most likely to eventually reach 

or exceed proficiency?

The projection model provides better predictions of who will 

eventually reach or exceed the proficiency cut score.

Under which model are interim growth expectations 

most clearly identified?

The trajectory and transition matrix models identify interim annual 

growth targets for each student. The projection model can be 

configured to specify targets for each student, but the process is 

less straightforward.
Evaluation of the Growth Model Pilot Project 58

Generalizing From a Standardized North Carolina Dataset

The application of three types of growth models to a common dataset requires a dataset with 

particular properties. Three are listed here, each corresponding to one particular type of model. 

First, for projection models, at least five years of longitudinal data are required in order to 

estimate a projection equation with a time horizon of three years. The first two years may be 

used as predictors, and the fifth year is used as the outcome. In practice, more predictor years 

are desired, but two years of predictors are used here for cross-model comparability. The North 

Carolina database is the only readily available dataset among the pilot states with sufficient years 

to estimate regression coefficients.

Second, the trajectory model requires a vertical scale or some method of mapping cross-grade 

scores onto a common scale. For present purposes the scores are standardized to the so-called 

“z-scale,” which has a mean of 0 and a standard deviation of 1 for each grade level. This is 

accomplished by taking each student’s score, subtracting the state’s average score in that 

student’s grade, and dividing by the state’s standard deviation of scores in that student’s grade. 

An implicit assumption of this method of scaling is that students should roughly maintain their zscores over time. Notably, this scaling approach allows for no substantive input about trends in 

student variation over time, no control over the expectation of whether high-scoring students 

should gain more than low-scoring students, and no input about whether student growth 

accelerates or decelerates over time. This atheoretical approach is thus inadvisable in practice

but useful as a “generic” scaling approach.

Third, transition matrix models require multiple cut scores below proficiency and, for growthonly models, above proficiency. In practice, the average state proficiency rate is around 65

percent, and the median state proficiency rate is around 70 percent. To set representative cut 

scores on the z-scale, a proficiency cut score of -0.5 for each grade is used. Assuming scores 

that follow a standard normal distribution, the proficiency rate would be 69.14 percent for each 

grade. For the transition matrix model, which requires multiple cut scores below proficiency, 

equally spaced cut scores on the z-scale of -0.8, -1.1, and -1.4 are set as seen in Exhibit 34. 

Exhibit 34

Illustration of Multiple Cut Scores for a Transition Matrix Model

4 Below 3 Below 2 Below 1 Below

Proficient and 

Above

In these four categories below proficiency, the percentages under normal distributions will be 9.7

percent, 7.6 percent, 5.5 percent, and 8.1 percent, for roughly equal proportions. These patterns 

are similar to those observed under transition matrix models in practice. Equally spaced cut 

scores are likely to be more generalizable than cut scores that generate equal proportions under 

normal assumptions, as there is no guarantee that normal assumptions will hold in practice or 

-1.4 -1.1 -0.8 -0.5
Evaluation of the Growth Model Pilot Project 59

over time. Equally spaced cut scores are also easy to extend above proficiency. This extension 

will be necessary for a growth-only version of the transition matrix model described later.

To reiterate, this standardized North Carolina dataset bears little resemblance to the original 

North Carolina dataset. Its cut scores are not North Carolina’s cut scores, and its trajectory 

model results will not follow the results reported in previous chapters. This was purposeful and 

results in a dataset with properties that generalize across states and support all three types of 

growth models. Certainly, growth model comparisons will be dependent on the scaling method 

and the cut scores in the data. When aggregating the student results to the school level for AYP 

purposes, systematic relationships between scales or cut scores and AYP results are also 

expected. Alternative cut scores than those shown in Exhibit 34 will be considered in a later 

section. More importantly, a later section introduces a general framework for visualizing the cutscore dependence of results.

Generic Growth Models

This section presents simplified, “generic” versions of growth models. These generic versions 

are designed to address two conflicting priorities: the comparability of results across models and 

the generalizability of results to actual state practices. To the extent possible, these models are 

designed so that differences can be interpreted as actual contrasts in model mechanics. To do 

this, cut scores, scaling decisions, and the available data are held constant across models while 

staying as true as possible to the operational models in growth model states.

This subsection presents status-plus-growth models, in which growth models have no effect on 

proficient students but classify some non-proficient students as “on-track.” A student under a 

status-plus-growth model is either proficient, non-proficient and on-track, or non-proficient and 

not on-track. A later subsection presents growth-only models, in which the proficient cut score 

only functions as a target, and a student is characterized as “on-track” or “not on-track” 

regardless of current proficiency status. 

The Generic Transition Matrix Model

The generic transition matrix model is similar to Delaware’s model in that it has four categories 

below proficient. It is similar to Iowa’s model in that any non-proficient student who has gained 

a category is considered on-track. Exhibit 35 displays the generic transition matrix model. 

Delaware’s model can be represented similarly by inserting fractions in the above-diagonal, nonproficient cells of Exhibit 35, but the generic model is simpler and gives uniform credit to any 

gains (i.e., the cells with ones). Two years of longitudinal data are required to locate students in 

the table presented in Exhibit 35.
Evaluation of the Growth Model Pilot Project 60

Exhibit 35

Student Weights Under the Generic Transition Matrix Model

Year t+1

4 Below 3 Below 2 Below 1 Below Proficient

Year t

4 Below 0 1 1 1 1

3 Below 0 0 1 1 1

2 Below 0 0 0 1 1

1 Below 0 0 0 0 1

Proficient 0 0 0 0 1

The Generic Trajectory Model

Trajectory models are used by Alaska, Arizona, Arkansas, Florida, and North Carolina and are 

described by Exhibit 9 in Chapter I. The generic trajectory model uses a linear trajectory from 

an examinee’s first-year score to a proficient cut score four years in the future. A non-proficient

examinee’s second-year score must be at least one-quarter of the distance to the proficient cut 

score in order to be designated as “on track.” Scores are standardized, and the proficient cut 

scores are set to be equal for all grades at -0.5. Thus, non-proficient students are on track if X2-

X1 ≥ (-.5-X1)/4, where X2-X1 is the gain over the first two years and -.5-X1

An equivalent expression of this model is that a student’s gains, extended linearly for three years, 

must reach proficiency. This expression would be X

is the distance from 

the year 1 score to proficiency. 

2 + 3(X2-X1

This generic trajectory model differs from the Arkansas model in that it is linear, but linearity is 

a simpler assumption and follows the other four trajectory model states. The generic model also 

differs from some models in that the time horizon from the initial year is four years. The effect 

of alternative time horizons is fairly straightforward to predict and is described in upcoming 

sections. Many state models have an “on-track in three years or by grade x, where x is 

determined by the state, whichever is sooner” policy. For the standardized North Carolina 

dataset with grades 3–8 and an assumed grade 8 graduation, this amounts to a policy where 

current eighth-graders must be proficient, current seventh-graders must be on-track in one year 

(N = 1), current sixth-graders must be on-track in two years (N = 2), and current fifth-graders and 

younger must be on-track in three years. The stricter time horizons for higher grades is a feature 

of the generic trajectory model.

) ≥ -.5, and, as expected, it is 

algebraically equivalent to the previous expression. Note that the first formulation establishes 

the initial year as the present-tense year with a four-year horizon, and the second formulation 

establishes the second, “current” year as the present-tense year with a three-year horizon. 

Because the latter formulation is more relevant to evaluating current year students, this model is 

described as an on-track-in-three model. Like the transition matrix model, this trajectory model 

requires only two years of data to evaluate students, as long as the cut score at the future time 

horizon is defined.
Evaluation of the Growth Model Pilot Project 61 

The Generic Projection Model 

The generic projection model is a simplified version of Ohio’s and Tennessee’s model. These 

simplifications achieve clear comparisons to the transition matrix and trajectory models. The 

simplified approach mimics the data use of transition matrix and trajectory models, which rely 

almost solely on the current year’s data (grade g) and the previous year’s data (grade g-1) to 

arrive at decisions. Similarly, for the simple projection model, students are treated as if they 

have only the most recent two years of reading and mathematics scores in order to make reading 

and mathematics on-track decisions, respectively, as seen in the following regression equations: 

R෡௚ାே ൌ Rഥ௚ାே ൅ ߚመ

ோଵ൫ܴ௚ିଵ െ Rഥ௚ିଵ൯൅ ߚመோଶ൫ܴ௚ െ Rഥ௚൯

 M෡௚ାே ൌ Mഥ௚ାே ൅ ߚመ

ெଵ൫ܯ௚ିଵ െ Mഥ௚ିଵ൯ ൅ ߚመெଶ൫ܯ ௚െ Mഥ௚൯. 

Here, R෡௚ାே and M෡௚ାே denote the predicted score for reading and mathematics in grade (g + N), 

respectively, ܰ years from the current grade ݃. Similarly, Ri and Mi denote the reading and 

mathematics scores in grade i, and ߚመ

௝௞ denotes the k

th regression coefficient for subject ݆. The 

regression coefficients cannot be estimated from the current cohort, as current students in grade 

݃ cannot know their scores in some future grade ሺ݃ ൅ ܰሻ. Thus, regression coefficients are 

estimated from a covariance matrix of test scores for a previous “reference” cohort that has 

completed the target grade level (g + N) of the state’s growth model. The prediction is obtained 

by entering current student scores into a prediction equation that uses the previous cohort’s 

regression coefficients. For these generic projection models, ܰ is set at 3, projecting to a grade 

three years from the current grade, following the on-track-in-three trajectory model. Due to the 

use of a standardized, “generic” dataset, these equations simplify as follows: 

ܴ෠

௚ାே ൌ ߚመ

ோଵ

መߚ ൅ ଵି௚ܴ כ

ோଶ

௚ܴ כ

ܯ௚෡ାே ൌ ߚመ

ெଵ

መߚ ൅ ଵି௚ܯ כ

ெଶ

௚ܯ כ

Here, the regression coefficients, ߚመ

௝௞

כ , are distinguished with a “כ “to denote their estimation 

from a past reference cohort. In practice, projection models typically use more than two 

predictors (Ohio and Tennessee, for example, require a minimum of three prior scores for 

predictions36). However, additional years of data would confound comparisons to transition 

matrix and trajectory models, as classification discrepancies could be explained by these 

additional data. Additionally, as upcoming sections argue, the conclusions drawn from the 

simplified projection model generalize to models with greater numbers of predictors. Students 

who are missing scores in the most recent two years in a particular subject will not be eligible for 

the growth model in that subject. This follows the same missing data approach as transition 

matrix- and trajectory-model states. 

 

36 These details are discussed at http://www2.ed.gov/admins/lead/account/growthmodel/oh/ohadd2.doc (p. 8) for 

Ohio, and at http://www.quickanded.com/2009/03/tennessee-growth-models-response-from.html (first footnote) for 

Tennessee. 
Evaluation of the Growth Model Pilot Project 62

An important difference between the generic projection model developed here and the projection 

model used in Ohio is that no confidence interval is applied to the generic projected scores. As 

described in Chapter II, Ohio estimated a standard error for each student’s projected score and 

added to the projected score a quantity equal to two standard error units; the augmented project 

score was then compared to the cut score for the target year in order to determine whether the 

student was on-track. The other pilot state using a projection model, Tennessee, did not adjust 

projected scores.

The available grades and years of the standardized North Carolina dataset are shown in Exhibit 

36. The most recent year with full data availability was 2006; this is established as the “current 

year” for student growth classification. For a hypothetical fourth-grade cohort in 2006, transition 

matrix and trajectory models need only two years of test score data, as represented by the central 

dark gray oval, to establish whether students are on-track. In contrast, projection models require 

a five-year dataset to project from grade 3 and grade 4 data to proficiency in the target grade 7 

(three years in the “future”), shown for the 2006 cohort as the gray oval in 2009. Because 2009 

data are not available in the “current year” of 2006, a reference cohort is used to estimate 

coefficients as described previously. This cohort is indicated in the table with the bolded arrow 

in the gray shaded cells. Once the coefficients are estimated from the reference cohort, the data 

from the current cohort are entered into the equation, and predicted student scores in grade 7 are 

compared to the grade 7 proficiency cut score to determine on-track status. This procedure is a 

simplification of Tennessee and Ohio’s approach that captures the fundamental regression-based 

features of the model while allowing contrasts to the other two model types.

Exhibit 36

Schematic Diagram of Data Required for Applying the Projection Model

2002 2003 2004 2005 2006 2007 2008 2009

Grade 3

Grade 4

Grade 5

Grade 6

Grade 7

Grade 8

Generic Model Extensions for Growth-Only Results

The generic trajectory model is extended easily to a “growth-only” approach by requiring that all 

students, whether proficient or not, be on track in three years or by a Grade 8 graduation. The 

same equations would apply to both proficient and non-proficient students. The projection 

model is likewise straightforward to extend: Both proficient and non-proficient students are 

entered into the prediction equation and must be predicted to be proficient in three years or by 

grade 8. 
Evaluation of the Growth Model Pilot Project 63

Transition matrix models require categories above the proficient cut score in order to extend 

inferences to proficient students who decline. A simple approach is to take the equally spaced 

cut scores below proficiency and extend them above proficiency. These cut scores follow, with 

the proficient cut score shaded:

-1.4 -1.1 -0.8 -0.5 -0.2 0.1 0.4 0.7 1.0 1.3

As before, a non-proficient student who does not change categories is not on-track. In contrast, 

the growth-only transition matrix model gives proficient students who do not change categories 

the benefit of the doubt and designates them as on-track. 

Student-Level Results for Status-Plus-Growth Models

As shown in Exhibit 36 above, the 2006 cohort (academic year 2005–06) is the most recent 

representative year of data in the available North Carolina dataset. The analysis focused on all 

Grade 3–7 students in this year that also had data from the previous year. These “two-year 

match rates” range from 90 percent to 96 percent. The results will be referenced by the grade of 

the students in 2006, the year of focus. The Grade 5 results, for example, refer to the students in 

Grade 5 in 2006 for whom Grade 4 data in 2005 were available. Grade 8 students are not 

eligible for the growth model, and their results are not reviewed. The 2006 Grade 3 Math cohort 

was not included because of extremely low match rates to the “Grade 2.5” pretest in that 

particular year.

As described above, cut scores were set in a generic fashion by standardizing scores and 

selecting a proficiency cut score of -0.5 on the z-scale. (See the “Alternative Cut Scores” section 

below for discussion of alternative proficiency cut scores.) A z-score of -0.5 results in about 69

percent of the students classified as proficient with normally distributed data. For reference, 

Exhibit 37 shows that resulting hypothetical proficiency rates based on the -0.5 cut scores hover 

around 70 percent as expected, with variability arising from deviations from normality and the 

discrete nature of the test scores.
Evaluation of the Growth Model Pilot Project 64

Exhibit 37

Overall Proficiency Rates for Reading and Mathematics for Each Grade in 2006 

Exhibit reads: Of the students assessed in 2006, 71.8 percent of those in Grade 3 are 

classified as proficient with respect to reading. Mathematics was not assessed for 

Grade 3. 

Exhibit 38 shows overall percentages of “on-track” students as classified by each model in each 

grade. The projection model always classifies fewer students as “on-track” and shows even 

greater relative stringency in math. For reading, the percentage of students classified as “ontrack” ranges from about 3.5 percent to about 8.5 percent, and for math, from about 3 percent to 

8 percent. The trajectory model becomes more stringent in Grade 6 and Grade 7, as is expected 

of a shortened time horizon to proficiency in Grade 8 (two years and one year respectively). 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Grade 3 Grade 4 Grade 5 Grade 6 Grade 7

Reading

Math
Evaluation of the Growth Model Pilot Project 65

Exhibit 38

“On-Track” Classification Rates for All Students for Reading and Mathematics 

by Model and Grade in 2006

Exhibit reads: Of the third-grade students assessed in reading in 2006, the generic 

transition model classified 5.9 percent as on-track, while 7.2 percent and 4.6 percent were 

classified as on-track by the generic trajectory and projection models, respectively.

Exhibit 39 shows the same results but, because growth models can only make a difference for 

non-proficient students when the results are used in the typical status-plus-growth framework, 

the percentages are rescaled to represent the classification rates for eligible (non-proficient) 

students. Whereas Exhibit 38 shows the percentage of all students classified as on-track by the 

growth model, Exhibit 39 shows the percentage of eligible (i.e., non-proficient) students 

classified as on-track by the growth model. Between 10 percent and 27 percent of eligible

students are classified as on-track by the three different growth models. The same pattern in 

relative magnitude among the three models as seen in Exhibit 38 is also apparent in Exhibit 39

for just the eligible students: The trajectory model tends to classify the most students as on0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

Grade 3 Grade 4 Grade 5 Grade 6 Grade 7

Reading

Transition

Trajectory

Projection

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

Grade 4 Grade 5 Grade 6 Grade 7

Mathematics
Evaluation of the Growth Model Pilot Project 66

track, followed by the transition matrix model. Within each grade, the projection model 

classifies the fewest eligible students as on-track. 

Exhibit 39

“On-Track” Classification Rates for Eligible (Non-Proficient) Students for Reading and 

Mathematics by Model and Grade in 2006

Exhibit reads: Of the eligible (non-proficient) students assessed in 2006, 21.0 percent of 

those in Grade 3 are classified students as “on-track” with respect to reading by the 

transition model.

It is noteworthy that the generic results are at odds with the empirical results shown in 

Exhibit 29, in which the two states using projection models had the highest rates of nonproficient students classified as on-track (75 percent in Ohio and 45 percent in Tennessee, 

compared with 25 percent or less in the other pilot states). As noted in Chapter II and in the 

section “Comparison of Student On-track and Proficiency Results” earlier in this chapter, much 

0%

5%

10%

15%

20%

25%

30%

Grade 3 Grade 4 Grade 5 Grade 6 Grade 7

Reading

Transition

Trajectory

Projection

0%

5%

10%

15%

20%

25%

30%

Grade 4 Grade 5 Grade 6 Grade 7

Mathematics

Transition

Trajectory

Projection
Evaluation of the Growth Model Pilot Project 67

of the high on-track rate observed in Ohio is likely explained by its procedure of augmenting the 

projected scores by two standard error units. However, an explanation for the high rate for 

Tennessee is not readily available. Two possible factors are listed here and discussed more fully 

in the upcoming sections. First, the model interacts with the difficulty of cut scores, both 

generally and across grades; this is discussed more fully below in the section “Alternative Cut 

Scores for Status-Plus-Growth Model.” As noted in that discussion, the projection model is 

particularly sensitive to where cut scores are located, with higher rates of non-proficient students 

identified as on-track when cut scores are lower. In 2007–08, Tennessee set relatively low cut 

scores and had one of the highest proficiency rates in the nation. Although these cut scores have 

changed, this may have contributed to the relatively high rate of on-track students observed in 

Exhibit 29. A second, less likely explanation is that the covariance structure of the Tennessee 

data differs dramatically from the covariance structure of the North Carolina data. However, 

cross-grade covariance relationships do not generally differ across states to an extent that 

explains the difference between Exhibits 29 and 39.

Exhibit 40 shows the pairwise percent agreement across models. Because all models classify 

proficient students as proficient, this is not counted as “agreement,” so only non-proficient 

students make up the denominator of the percent agreement statistic. The findings suggest that 

the transition matrix and trajectory models have similar mechanisms and identify greater 

proportions of students as “on-track” than projection models. Transition matrix and trajectory 

models agree on over 90 percent of growth-eligible (non-proficient) students. Their agreement 

rates with projection models are closer to 60 percent. This agreement is wholly accounted for by 

common classification of not on-track students, with the exceptions of Grade 4 Reading and 

Grade 7 Math, which have a handful of students who are commonly classified as on-track. 

Agreement about on-track students is near zero. That is, if a transition matrix or trajectory model 

classifies a student as making adequate growth, the projection model almost always disagrees, 

and vice versa. The theoretical reasons for this are laid out in the section below, entitled “A 

Framework for Comparing the Models.”
Evaluation of the Growth Model Pilot Project 68

Exhibit 40

Pairwise Percent Agreement of Models for Eligible (Non-Proficient) Students,

Reading and Mathematics

Exhibit reads: The transition matrix (Trans) and trajectory (Traj) models agreed on 94.3 percent 

of eligible (non-proficient) Grade 3 students assessed with respect to reading. The transition 

matrix and projection (Proj) models agreed on 62.6 percent, while the trajectory and projection 

models agreed on 58.1 percent in reading.

As noted previously, the generic projection model differs more from its real-life counterpart than 

the transition matrix and trajectory models in that this projection model uses only one subject 

and only two years of prediction. This was done to ensure that all growth models were using 

similar data. The projection model can be extended across subjects and grades, but this will not 

necessarily increase the agreement rates. The reasons for this are also laid out in the section “A 

Framework for Comparing the Models” below.

Student-Level Results for Growth-Only Models

This section overviews the results from a growth-only approach. This parallels the analyses at 

the beginning of this chapter in which growth-only criteria were applied to each GMPP state. 

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Evaluation of the Growth Model Pilot Project 69

The results here also parallel the immediately prior status-plus-growth section, in which all 

models agreed about proficient students, leaving model contrasts centered on non-proficient

students. In this section, students are either on-track or not on-track, and models can 

theoretically disagree about any student. 

Exhibit 41, below, is the analog of Exhibit 39 from the status-plus-growth analysis. Exhibit 41

shows the “on-track” classification rates for all students by model and grade. Transition matrix 

models classify the fewest students as on-track: between 53 percent and 57 percent. Trajectory 

models classify more students as on-track: between 59 percent and 68 percent. The higher 

percentages are in the higher grades. The two-year horizon in Grade 6 and the one-year horizon 

in Grade 7 are less lenient for non-proficient students but more lenient for proficient students. 

Because there are more proficient students, this manifests as an increase in the overall on-track 

classification rate. Projection models have the highest classification rates: between 70 percent

and 74 percent and fairly uniform across grades.

Exhibit 41

“On-Track” Classification Rates Using Growth-Only Results for All Students for Reading 

and Mathematics, by Model and Grade in 2006

Exhibit reads: Of the students assessed in 2006, 54.0 percent of those in Grade 3 are classified 

as “on-track” with respect to reading by the transition model.

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Evaluation of the Growth Model Pilot Project 70

Exhibit 42 shows the pairwise percent agreement of the models, defined as the percentage of 

students for whom on-track classifications are agreed upon by a pair of models. This is the 

analog of Exhibit 40 from the status-plus-growth analysis. Transition matrix and trajectory 

models are again the most similar in overall function, hovering around 90 percent agreement for 

the lower grades and decreasing in the higher grades to near 82 percent. This decline in the 

higher grades is explained by the decreasing time horizon and can be visualized in the upcoming

section “A Framework for Comparing the Models.” 

Exhibit 42

Pairwise Percent Agreement of Models Using Growth-Only Results for All Students in 

Reading and Mathematics in 2006

Exhibit reads: The transition matrix (Trans) and trajectory (Traj) models agreed on 

92.1 percent of Grade 3 students assessed with respect to reading.

Exhibit 42 also shows that the lowest agreement in each grade is between the transition matrix 

and projection models: between 62 percent and 69 percent. Exhibit 42 shows similarly low 

agreement between the trajectory model and the projection model. The agreement rates are 65

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Evaluation of the Growth Model Pilot Project 71

percent, but they increase to 80 percent and 85 percent in Grade 7. This dramatic increase in 

agreement is largely explained by the increased leniency of the trajectory model on proficient 

students due to the time horizon. The explanatory framework developed below illustrates why 

this agreement between the trajectory and projection models increases with convergence toward 

the end of the growth model’s grade span. 

Alternative Cut Scores for Status-Plus-Growth Models

One important way in which states differ is in the setting of cut scores, with some states setting 

higher cut scores and typically posting lower proficiency rates while others set lower cut scores 

and classify more students as proficient. In this section, we briefly examine results of using the 

same data but raising the proficiency cut score from -0.50 to -0.25. For these data, this results in 

proficiency rates near 60 percent compared to the previous proficiency rates nearer to 69 percent. 

To adapt to this change, the transition matrix model cut scores below proficiency were also 

raised slightly and spaced at 0.35 intervals instead of 0.3.

Exhibit 43 shows that transition matrix and trajectory models classify larger proportions of 

students (compare to Exhibit 38), because there are more non-proficient students to classify. In 

contrast, the projection model classifies slightly fewer students. The raised cut scores leave more 

non-proficient students to classify, but the classification approach of projection models, which 

contrasts so strongly with the other two models, interacts differently with the bivariate 

distribution of scores. This can be visualized in the upcoming section, “A Framework for 

Comparing the Models.”37 

 37 The cut-score dependence of growth models has been described by A.D. Ho, D.M. Lewis, and J.L.M. Farris 

(2009) The dependence of growth-model results on proficiency cut scores. Educational Measurement: Issues and 

Practice, 28 (4), 15–26.
Evaluation of the Growth Model Pilot Project 72

Exhibit 43

“On-Track” Classification Rates for All Students for Reading and Mathematics by Model 

and Grade in 2006: Higher Cut Scores, Proficiency Rates Near 60 Percent

Exhibit reads: Of the students assessed in 2006, 8.4 percent of those in Grade 3 are 

classified as “on-track” with respect to reading by the transition model when higher cut 

scores are used.

Exhibit 44 shows the proportion of eligible students classified by the models. These values are 

generally similar to Exhibit 39 with the exception of the projection model identification rates, 

which have declined substantially. As cut scores increase and proficiency rates decrease, the 

relative impact of projection models will decline. This is also visible in the upcoming 

framework in the following section.

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Evaluation of the Growth Model Pilot Project 73

Exhibit 44

“On-Track” Classification Rates for Eligible (Non-Proficient) Students for Reading and 

Mathematics by Model and Grade in 2006: Higher Cut Scores, Proficiency Rates Near 60 

Percent

Exhibit reads: Of the eligible (non-proficient) students assessed in 2006, 23.6 percent 

of those in Grade 3 are classified as “on-track” with respect to reading by the transition 

model when higher cut scores are used.

A Framework for Comparing the Models

Differences and disagreements in growth model classification can be visualized with a bivariate 

framework (modified from Ho, Lewis, and Farris, 2009). The framework allows visualization of 

the kinds of students classified as on-track by each of the three models and explains the bar-plot 

exhibits from the preceding sections in a more coherent structure. The framework is somewhat 

complex, but it affords the following:

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Evaluation of the Growth Model Pilot Project 74

• Visualization of the transition matrix model as a categorical approximation of the 

trajectory model.

• Visualization of the projection model as a stark contrast to the trajectory and transition 

matrix models.

• Visualization of the projection model as a model that rewards high average achievers, 

whereas the trajectory and transition matrix models can be visualized as models that 

reward students making gains.

• Visualization of the projection model’s relative preference for previously high-scoring 

decliners and both the transition matrix and trajectory models’ relative preferences for 

low-scoring gainers, albeit those that might be less likely to actually reach proficiency.

• Visualization of the projection model’s relatively low classification rates for status-plusgrowth models but relatively high classification rates for growth-only models.

• Visualization of how the trajectory model’s classification rate over the projection model 

will increase with increasing cut scores.

• Visualization of how shorter time horizons increase the overlap between trajectory and 

projection models.

Exhibit 45, on the following page, displays lines that overlay an imagined scatterplot of student 

scores, with a current year’s scores plotted against a past year’s scores. The scores are not 

shown to prevent a cluttering of the diagram, but they may be imagined as a cloud centered on 

the point (0,0) in the upper right portion of the plot and stretching as an ellipse from the lower 

left to the upper right due to the correlated nature of adjacent-grade scores. In this context, the 

y-axis represents the 2006 scores for a given grade cohort, say, Grade 5, and the x-axis 

represents the 2005 scores for the same cohort, in this case, in Grade 4. The scores are 

standardized within both years to a mean of 0 and a standard deviation of 1, so axis values can be 

interpreted on the z-scale. The framework intentionally focuses on lower-scoring students, and 

the figure is bounded by -2 and 0.5. Important features of the plot include the main diagonal, 

along which students have the same z-scores in both years. Students above this diagonal are 

increasing their scores (in standard deviation units from the mean) over time, and students below 

this diagonal are declining. The proficiency cut scores are set at z-scores of -0.5 in both 2005 

and 2006, as described above. These are represented by a bold vertical and horizontal line 

respectively. Students above the horizontal “Proficient in 2006” line are proficient in the 

“current year” and are unaffected by status-plus-growth models.

The transition matrix model cut scores of -0.8, -1.1, and -1.4 are also shown.38

 38 The consequences of higher cut scores are generally predicted by the framework, which slides up the diagonal 

over the imagined bivariate scatterplot in Exhibit 45. The target areas illustrated by Exhibit 45 begin to incorporate 

more central and denser regions of the bivariate scatterplot.

 Under the 

transition matrix model, non-proficient students who have gained a category are deemed “ontrack.” This can be visualized in Exhibit 45 by the shaded regions. All of the shaded regions lie 

above the main diagonal, as befits any increase in z-scores, and only non-proficient students are 

highlighted, as those above the -0.5 line are simply proficient. The staircase shape arises from 
Evaluation of the Growth Model Pilot Project 75

the categorical nature of the transition matrix model, which does not distinguish score levels 

within any score category.

Exhibit 45

A Bivariate Framework Showing the Non-Proficient Students Deemed

“On-Track” by a Transition Matrix Model

Exhibit reads: The shaded area corresponds to students who were not proficient in 2006 but 

whose growth from 2005 to 2006 classified them as on-track under the transition matrix model.

Non-proficient students in 2006 classified as on-track by the trajectory model lie in the triangle 

(not shaded) above the Trajectory Model line and below the Proficient in 2006 line. Nonproficient students in 2006 classified as on-track by the projection model lie in the triangle (not 

shaded) above the Projection Model line and below the Proficient in 2006 line.

Exhibit 45 also includes lines for the other growth models. The trajectory model line arises from 

the trajectory model decision rule: If a student’s score gains from 2005 (x) to 2006 (y) are 

continued for the next three years (or to eighth grade, whichever is sooner) and the student is 

then proficient or above, then that student is on-track. On this scatterplot, this amounts to the 

following equation, which is again algebraically equivalent to the generic trajectory model 

formulation presented in the previous section: 

If y < -0.5 (non-proficient) AND y + 3*(y-x) ≥ -0.5, then on-track. 

These equations are both plotted as lines in Exhibit 45: the Proficient in 2006 line and the 

Trajectory Model line respectively. Students classified as “on-track” by the trajectory model will 

lie in the semi-infinite triangle (not shaded) above the Trajectory Model line and below the 

Proficient in 2006 line.
Evaluation of the Growth Model Pilot Project 76

For later grades, the proficiency horizon may be less than three years, so the slope of the 

trajectory model line will decrease and pivot about the “proficiency origin” of (-0.5, -0.5). If all 

else is equal across grades, this will effectively decrease the proportion of students classified as 

“on-track” for Grade 6 students (who must be on-track in two years) and Grade 7 students (who 

must be on-track in one year). This decline can be seen in Exhibits 38 and 39.

Exhibit 46 shows areas of discrepancy between transition matrix and trajectory models. Students 

falling in the darker-shaded triangles will be classified as on-track by the trajectory model but 

not the transition matrix model. Students falling in the lighter-shaded triangles will be classified 

as on-track by the transition matrix model but not the trajectory model. These triangles are fairly 

small when all non-proficient students below the Proficient in 2006 line are considered. This

accounts for the high agreement rates for the two models in Exhibit 40. Discrepant 

classifications are fairly balanced in this case.

Exhibit 46 uses Grade 5 data for an “on-track in three years” model. For Grades 6 and 7, in 

which the slope of the Trajectory Model line decreases, the discrepant classifications will be less 

balanced. Not only will the trajectory model begin to classify fewer students as on-track, but the 

discrepancies will begin to be more fully accounted for by the leniency of the transition matrix

model. Exhibit 46 illustrates that the conceptual simplicity of the transition matrix models’

discrete performance categories carries a cost of some misclassification of students compared 

with the trajectory model. However, the interpretability of the transition matrix model’s studentlevel results may be more straightforward for teachers, students, and parents than score-based 

trajectory models despite the loss in classification accuracy. 
Evaluation of the Growth Model Pilot Project 77

Exhibit 46

Areas of Discrepancy Between Transition Matrix and Trajectory Models

Exhibit reads: The dark-shaded areas correspond to students classified as on-track by the trajectory 

model but not by the transition matrix model. The light-shaded areas correspond to students 

classified as on-track by the transition matrix model but not by the trajectory model.

The generic projection model decision rule is illustrated by the line with negative slope labeled 

“Projection Model” in Exhibit 46. The derivation of this line is explained in Appendix D. 

Exhibit 47 removes the transition matrix model borders to unclutter the diagram and focuses on 

the classification area for the projection model. The projection model will classify students who 

are currently (2006 in this illustration) below proficient as on-track if their scores are in the 

shaded area in Exhibit 47. The near-perpendicular relationship of the projection model line to 

the trajectory model line helps to explain the reason for the low percent agreement shown in 

Exhibit 40.

What is striking about this area is its lack of overlap with the on-track areas for the other two 

models. The vast majority of students classified as on-track by the projection model are in fact 

below the main diagonal: they are showing score declines over time, and most were proficient in 

2005 but not proficient in 2006. This is a dramatic departure from the logic of transition matrix 

and trajectory models, but it is not so surprising in the context of regression. When scores have 

high and similar correlations across years, as is typical of educational test scores, the slopes of 

projection model lines will be negative (see Appendix D). This translates to the statement, high 

past scores predict high future scores. Projection models are actually insensitive to the 

chronological order of grades from which the prediction is made. For example, if a third-grade

score is very high, projection models allow that to compensate for a fourth-grade score that is 
Evaluation of the Growth Model Pilot Project 78

below proficient. This feature of projection models is unchanged by increasing the numbers of 

years and subjects for prediction; this would increase the accuracy of prediction but not change 

the nature of the model. 

Exhibit 47

The Projection Model Classification Approach

Exhibit reads: The shaded area corresponds to students classified as on-track by the projection model.

Exhibit 47 also allows contextualization of the percent agreement rates in Exhibits 40 and 42. 

All models agree that students in the tent shaped triangle under the main diagonal and the 

Projection Model line are not on-track. In this case, however, there is not one single student 

whom all three models classify as on-track. The projection model area and the trajectory model 

area only overlap in the small triangle in the center of the figure, bounded by the Projection 

Model line, the Trajectory Model line, and the Proficient in 2006 line. In the real data example 

for the Grade 5 Math cohort in 2006, due to the discrete nature of test scores, there are no 

students in this triangle at all. In this case, all positive projection model classifications are not 

supported by the other two models, and vice versa.

This stark contrast represents two competing views about the prediction of progress, one 

educational and one statistical. The educational view is based on a momentum metaphor, 

ironically both a trajectory and a projection: objects in motion will remain in motion. The 

statistical view takes the chronological ordering of events into account only as much as proximal 

correlations are greater than distal correlations and, through regression techniques, offers a 

statistical prediction that is effective but blind to progress over time.
Evaluation of the Growth Model Pilot Project 79

Exhibit 46 also explains why projection model classification rates are lower than others. Student 

scores are not uniformly distributed throughout the plot. More students hover close to the 

diagonal due to the positive correlation of adjacent-grade scores. As a result, more students will 

be located in the trajectory-area triangle than in the projection-area triangle. This leads to the 

findings in Exhibit 38. Further consideration of the framework with respect to the underlying 

bivariate distribution allows for visual explanations of all the bar graphs shown in previous 

exhibits.

Extending the Framework to the Growth-Only Results

The previous sections applied the generic growth models to a single, common dataset using the 

status-plus-growth rule as is typically done in the GMPP states. This section extends that 

analysis by applying the hypothetical “growth-only” rule when employing the various generic 

growth models to the common dataset. This extension is similar to the application of the 

“growth-only” rule to the GMPP states in the sections at the beginning of this chapter.

Extensions of the status-plus-growth framework to include use of on-track indictors for all (nonproficient and proficient) students are straightforward and are shown here. Exhibit 48 adds 

transition matrix model categories above the proficient cut score and shows the growth-only 

transition matrix model approach. Category maintenance below proficiency is classified as not 

on-track and category maintenance above proficiency is classified as on-track. The 90 percent

agreement rate with the trajectory model is unsurprising here, as the cutoff lines for the transition 

matrix model zigzag across the cutoff line for the trajectory model.
Evaluation of the Growth Model Pilot Project 80

Exhibit 48

Transition Matrix Model Classification Areas for On-Track Students in 2006

Exhibit 49 shows the contrasting areas of on-track classification for trajectory and projection 

models under a growth-only approach. The dark-shaded area on the left shows the area where 

students who score low in the first year but make gains are recognized as on-track by the 

trajectory model but overlooked by the projection model. The medium-shaded area on the right 

shows the area where students who score high in the first year but make declines are recognized 

as on-track by the projection model but passed over by the trajectory model. This figure also 

makes it clear that decreasing the slope of the Trajectory Model line will increase the agreement 

between the two models. This is exactly what happens as the time horizon decreases in Grades 6 

and 7, and this explains the increasing agreement between trajectory and projection models in 

those grades.
Evaluation of the Growth Model Pilot Project 81

Exhibit 49

Areas Where Trajectory Models and Projection Models Will Disagree

Finally, Exhibit 49 explains why projection models classify relatively fewer students in statusplus-growth models but relatively more students in growth-only models. Due to the location of 

the bivariate scatterplot, centered on (0,0) and stretching along the diagonal, the projection 

model’s shaded area (including both the medium- and light-shaded regions) simply captures 

more students. Importantly, the framework allows for the visualization of the cut-score 

dependence of this finding. If the cut scores were higher, the trajectory model line would drag 

up the diagonal, but the projection model line would move straight up (changing only the 

intercept), thus the projection model and trajectory model classification rates would both decline 

but become more similar.

School AYP Simulations Based on the Generic Models

The student-level analyses presented above indicate that the models yield different rates of 

students classified as on-track to proficiency, and that the students so identified are quite 

different in the projection model than with the transition matrix and trajectory models. The last 

step of the generic model comparison is to aggregate the student results to the school level in 

order to simulate AYP determinations.

Two methods of determining AYP are used here. The first compares closely to the method used 

in all of the pilot states except Delaware, what was referred to as “status-plus-growth” in 

Chapter I. This involves first calculating the percentage of students in the school who are 

proficient in both reading and mathematics. If this percentage meets or exceeds the annual 
Evaluation of the Growth Model Pilot Project 82

measurable objective (AMO), the school made AYP and is classified as having made it by status. 

If the school did not make AYP by status, then the on-track data from the growth model are used 

for all non-proficient students. If the addition of the on-track data to the proficient data raises the 

school percentage to meet or exceed the AMO, the school is classified as having made AYP by 

growth. 

The second method parallels the procedure that Delaware followed, whereby the growth model 

on-track data are used first. If the percentage on-track exceeds the AMO, the school made AYP 

by growth. If the percentage on-track was less than the AMO, then students who were proficient 

but not on-track to maintain proficiency are added to the on-track numbers. If that percentage 

meets or exceeds the AMO, the school is classified as making AYP by growth-plus-status.

Both methods require setting subgroup minimum size requirements, and we selected 40 students 

as the minimum. The proficiency cut score level used in this simulation is the -0.50 level used in 

the initial student-level analysis.

The results for the first (status-plus-growth) method are shown in Exhibit 50. Using the -0.50 cut 

point for proficiency and the average AMO levels for North Carolina in 2006–07 (56 percent), 

only about 20 percent of the schools met the AMO and would make AYP with the status model. 

To determine the number of additional schools that would make AYP using the growth data, the 

growth model results for non-proficient students in subgroups that did not meet the AMO are 

then added to the number of proficient students in those subgroups, and the pass rate was 

recalculated. If all percentages meet the AMO, the school is classified as making AYP by 

growth. The trajectory model yielded a slightly higher percentage, about 15 percent, than the 

projection and transition matrix models. These results are consistent with the student level 

findings reported earlier, that the trajectory model generally identifies a higher proportion of 

non-proficient students as on-track to proficiency than the other models. 

Exhibit 50

Percentage of Schools Making AYP by Status and by Growth Based on Results 

from the Different Types of Generic Growth Models

Exhibit reads: In addition to the 20 percent of schools making AYP by status, the trajectory 

model classifies another 14 percent of schools as making AYP.

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Evaluation of the Growth Model Pilot Project 83

The AYP results from the different types of models change markedly if the growth model results 

are given priority and schools are first evaluated on whether they meet the AMO using growthonly results. Exhibit 51 shows that only 3 percent of the schools made AYP using the trajectory 

model to classify students as on-track to proficiency and then comparing the percentages of 

students on-track with the AMO. The transition matrix model yielded similar results but with 

even fewer schools making AYP by growth only. In contrast, the projection model results in 

about 30 percent of the schools making AYP by growth only.

This is consistent with the student level results and the theoretical framework laid out earlier, 

both of which revealed the tendency of the projection model to classify more students as on-track 

to maintain or exceed proficiency in a growth-only context. As the previous section described, 

the projection model tends to operationalize student growth more as high average scores. With 

moderately low (but realistic) cut scores of -0.5, there are many high average scorers. In 

contrast, trajectory and transition matrix models require growth in a growth-only context, and 

there are many decliners, whether proficient or not. This discrepancy results in projection 

models having a relatively large impact in a growth-only setting but a relatively small impact in a 

status-plus-growth setting at both the student and school levels.

Exhibit 51

Percentage of Schools Making AYP by Growth-Only and Growth-Plus-Status Based on 

Results from the Different Types of Generic Growth Models.

Exhibit reads: The trajectory model classifies 3 percent of schools as making AYP by 

growth only and an additional 32 percent of schools as making AYP if status criteria are 

added after growth is applied.

Consistency of AYP Determinations With Future Student Learning

This analysis builds on results of previous comparisons of generic models to assess the predictive 

accuracy of growth model results at the student level. To compare correct-classification rates, a 

full replication of all analyses was performed, substituting a 2003 year of focus for the original 

2006 year of focus. The numbers of students and the match rates were smaller, but results did 

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Evaluation of the Growth Model Pilot Project 84

not differ substantively from the 2006 results. Unlike the 2006 year of focus, which was the 

most recent year of representative data availability, the 2003 year of focus allows a later validity 

check for 2003 classifications. Grades 3–5 on-track predictions could be evaluated using the 

2006 data, three years later, and Grade 6 and 7 predictions could be evaluated two years and one

year later in 2005 and 2004 respectively. 

Exhibit 52 shows the correct-classification rates by model and grade for a growth-only model: 

how often on-track students were actually proficient and not-on-track students were actually nonproficient. As expected of a regression model, the projection model correct-classification rate is 

higher, by around 20 percentage points over the transition matrix and trajectory models in the 

lower grades. As the time horizon shrinks in Grades 6 and 7, and the trajectory model 

classification line more closely approaches the projection model’s classification line, the 

projection model advantage drops under 10 percentage points.
Evaluation of the Growth Model Pilot Project 85

Exhibit 52

Correct Classification Rates by Model and Grade, Reading and Mathematics

Exhibit reads: Of the third-graders who were classified with the generic transition model 

as on-track or off-track in reading, 59.9 percent turned out to achieve their predicted 

status of proficiency or non-proficiency. 

Whereas Exhibit 52 shows correct classifications for both on-track and non-on-track students, 

Exhibit 53 reports the percentage of on-track-designated students that actually reach proficiency. 

Naturally, these percentages are higher, with the projection model’s advantage decreasing from 7 

percentage points in the lower grades to less than 5 percentage points in the higher grades. The 

ranking of the models is the same across all subjects and grades.

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Evaluation of the Growth Model Pilot Project 86

Exhibit 53

Percentage of On-Track-Classified Students Who Achieve Proficiency by Model and 

Grade, Reading and Mathematics

Exhibit reads: In reading, 75.8 percent of the Grade 3 students that the generic transition 

model classified as on-track turned out to achieve proficiency.

However, trajectory models may have utility through appealing to the logic of teachers and 

students in schools. Students identified by the trajectory model may indeed make a gain due to 

measurement error, but it may make more educational sense to teachers and administrators to 

reward observed gains than high-averaging declines. These models thus contrast both in setting 

incentives and describing the impact of policies. These models establish both predictions and 

examples for the kinds of students that deserve recognition. Given these tradeoffs, states may 

choose their balance of priorities as they develop future growth models. 

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Evaluation of the Growth Model Pilot Project 87

Comparison of Generic Growth Models: Summary and Conclusions

The projection model functions in stark contrast with transition matrix and trajectory models in 

terms of how they classify students as on-track and not no-track. When a projection model 

classifies a non-proficient student as on-track, the probability that a transition matrix or trajectory 

model agrees is near zero, and vice versa. Agreement rates for non-proficient students are 

around 60 percent due to agreement about students not on-track.

For status-plus-growth models (in which only non-proficient students are classified as on-track 

or not on-track), projection models will have the least impact, affecting only 10–20 percent of 

eligible (non-proficient) students while transition matrix and trajectory models affect over 20

percent. States with higher cut scores will have heightened differences between the model types 

and a lower proportional impact of projection models on eligible students.

For growth-only models, in which all students are classified either as on-track or not on-track, 

projection models will be the most lenient, classifying around 70 percent of students as on-track

for typical cut scores, around 10 percentage points greater than the other models.

A graphical framework allows for visualization of the differences between the models. It also 

affords clear predictions under alternative assumptions about distributions and cut scores. 

Using the data from the generic models to simulate school AYP determinations, the models do 

not yield large differences in the percentages of schools making AYP for status-plus-growth 

models. The models differ much more under a growth-only approach, in which very few schools 

would make AYP with a trajectory or transition matrix model, and many schools would make 

AYP with a projection model.

The underlying reasons for the contrast between the projection model and the transition matrix or 

trajectory models relate to differing objectives guiding the models. Projection models are 

designed to maximize the accuracy of a prediction about whether a student will meet or exceed 

future proficiency standards. From a statistical standpoint, the best predictor of future 

achievement is past achievement. As a result, relatively few students with records of low 

achievement are predicted to meet or exceed future proficiency standards, while students with 

records of high achievement are very likely to be predicted to do so. Transition matrix and 

trajectory models, in contrast, are designed to identify specific growth targets that each student 

must attain in order to meet or exceed the future proficiency standards, given his or her 

benchmark score (usually from the last test administration). Reflecting these different guiding 

objectives, projection models classify as on-track a large number of previously proficient 

decliners (many of them correctly), whereas transition matrix and trajectory models classify as 

on-track a large number of lower-scoring gainers (some of them incorrectly). The educational 

significance of this contrast is briefly discussed.
Evaluation of the Growth Model Pilot Project 88

Effects of Different Growth Standards for School Accountability

The previous sections compared three types of growth models. According to the GMPP policy, 

each of these follows the bright-line principle of ensuring universal proficiency by 2014. Thus, 

these models set individual growth standards quantifying adequate progress to proficiency by a 

time horizon, graduation, or 2014: These are growth-to-proficiency models. Alternative models 

exist that do not depend or depend less on proficiency standards and time horizons. These 

models may define adequate growth for each grade or different starting proficiencies, or a 

common growth standard that can be applied to all students. It is important to note that such 

alternatives are not currently allowed for use in ESEA-mandated AYP reporting. Nonetheless, it 

is potentially useful to make controlled comparisons between growth-to-proficiency models and 

alternative types of growth models.

The analysis that follows assesses the hypothetical impact on school AYP determinations of 

using a different standard of “adequate yearly growth” than the growth-to-proficiency piloted 

under the GMPP. Toward this end, we constructed a growth indicator based on the difference 

between the proficiency cut points of successive grade levels. Non-proficient students who gain 

more than that difference are classified as on-track, even though their growth trajectory may not 

be steep enough to reach proficiency within the three- or four-year time frame of the growth 

model. For this analysis, the data are drawn from the actual Florida database (along with the 

actual Florida proficiency cut scores) and include students with test scores in 2007–08 and 2006–

07. Florida’s data are well suited for this simulation because the state provided longitudinally

linked scale scores for their students and the scale scores were vertically equated.

A comparison of results from the actual GMPP growth-to-proficiency model with the alternative 

growth standard is shown in Exhibit 54. The upper panel shows the actual GMPP and the lower 

panel shows the alternative growth standard results. The first vertical bar for each grade level in 

both panels is the percentage of students that were proficient or higher in both reading and 

mathematics at each grade level; these ranged from 48 to 62 percent across these grade levels. 

The second bar (labeled “GMPP on track” in the upper panel and “alt. growth” in the lower) is 

the percentage of students who were classified as making “adequate growth” under the 

respective growth models. The percentages of students who were on-track per the GMPP growth 

model (upper panel) ranged from about 44 to 52 percent across grades. In contrast, the 

percentages that met the alternative growth standard (i.e., that gained an amount equal to or 

greater than the difference between their current grade level proficiency cut score and the 

proficiency cut score for the prior grade level) were much lower, ranging from 16 to 23 percent 

except for grade 7 where 38 percent of students were proficient under the alternative growth 

standard. The large increase in grade 7 was likely a reflection of an unusually small distance 

between the proficiency cut scores for grade 6 and grade 7.
Evaluation of the Growth Model Pilot Project 89

Exhibit 54

Percentages of Students Scoring At or Above Proficiency, Making GMPP Annual Growth 

Targets (Upper Panel), Making Alternative Annual Growth Standard (Lower Panel), and 

Meeting Either Proficiency or Growth Target, Florida Data from 2006–07 and 2007–08 

(Upper Panel: GMPP Growth)

Exhibit reads: In 2007–08, 62 percent of the grade 4 students in Florida scored at or above the 

proficiency cut scores in both reading and mathematics. Fifty percent of the students were classified 

as on-track by the Florida GMPP growth model, and 73 percent were either proficient or on-track.

(Lower Panel: Alternative Growth)

Exhibit reads: In 2007–08, 62 percent of the grade 4 students in Florida scored at or above the 

proficiency cut scores in both reading and mathematics. Sixteen percent of the students were 

classified as making adequate annual growth by the hypothetical alternative growth model, and 

70 percent were either proficient or making adequate growth.

This simulation shows that the percentages of students meeting this alternative standard of 

growth are lower than the percentage of proficient students in both reading and mathematics. 

0

10

20

30

40

50

60

70

80

Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 All Grades

Status

GMPP 

on-track

Status 

plus 

GMPP 

on-track

0

10

20

30

40

50

60

70

80

Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 All Grades

Status

Alternative 

on-track

Status plus 

alternative 

on-track
Evaluation of the Growth Model Pilot Project 90

However, adding non-proficient students who met the growth standard to the pool of proficient 

students would increase the overall rates of students who are arguably performing adequately 

(i.e., proficient or making reasonable progress over the past year).

A possible shortcoming of this method of assessing adequate yearly growth is that it may be 

negatively correlated with initial achievement level, such that students with higher initial 

achievement gain less and are more likely to not make adequate yearly growth by this standard. 

This could occur because the state tests are designed to measure achievement with respect to 

grade level standards and may be prone to ceiling effects. If the model is less sensitive to gains 

made by higher-achieving students, the minimally acceptable growth increment could be 

conditioned on initial level of achievement.

Exhibit 55 shows evidence consistent with ceiling effects in reading and mathematics in Florida. 

This graph plots the average scale-score gains from 2006–07 to 2007–08 on the vertical scale (yaxis) by the decile of the student’s 2006–07 score. The average gains in both subjects are 

greatest among students with the lowest scores in 2006–07. The average gains gradually decline 

from decile 2 to decile 8 and then drop more markedly in decile 10. These results suggest that, at 

least for this assessment system, somewhat different approaches to assessing adequate yearly 

growth may be required if a growth-only model is being considered. In a status-plus-growth 

model, however, lower expected growth is unlikely to have an impact on how high-achieving 

students are classified, because these students are proficient and the growth model is only 

applied to non-proficient students.

Exhibit 55

Average One-Year Gains in Mathematics and Reading, by Base Year Achievement Decile

Exhibit reads: Florida students who scored in the lowest decile had average one-year gains 

of 260 scale-score points in mathematics and 300 scale-score points in reading.

This alternative growth standard is similar to some state (as opposed to federal) accountability 

systems in that it is not grounded in students being on-track to meet or exceed proficiency cut 

-50

0

50

100

150

200

250

300

1 2 3 4 5 6 7 8 9 10

Scale-score Gain 

from 2006-07 

to 2007-08

2006-07 Achievement Decile

Math

Reading
Evaluation of the Growth Model Pilot Project 91

scores. However, this hypothetical alternative probably yields lower rates of students and 

schools classified as making adequate growth than the growth models used in most state 

accountability systems. State accountability growth models often define adequate growth as 

“any growth” while this hypothetical formulation defines adequate growth in terms of a specific 

common yardstick for each grade.

Effects of Longitudinal Matching

The final aspect of growth models examined here is the extent to which the state assessment 

systems were able to calculate growth indicators for their students. Growth indicators are 

generally based on at least two years of achievement data. However, some students in each 

school year will not have prior test score data because they transferred in from out of state or 

from a private school, were absent on the test dates, or were excluded from the prior year testing 

for one reason or another. The percentage of students with growth indicators is referred to as the 

match rate (all of the pilot states except Ohio and Tennessee required two consecutive years of 

test scores in order to calculate growth indicators). 

ESEA requires that schools test each year at least 95 percent of their students in both 

mathematics and reading or language arts. However, it is possible for the percentage of students 

with test scores from both the current and the prior year (students whose scores can be 

‘matched’) to be lower than 95 percent.

The match rate has potentially important implications for the validity of a growth model: if a 

growth-only model is applied to all students in an ESEA reporting group in order to assess 

whether the AMO was met, then only the matched students are used to determine proficiency 

rates. Among the nine states examined here, only Delaware, Florida, Ohio, and Tennessee 

applied their growth models to all students in ESEA reporting groups. In Delaware, growth 

model results were considered first in making AYP decisions for each ESEA reporting group. 

Florida, Ohio, and Tennessee applied growth results after status and safe-harbor but used the 

growth model results for all students within ESEA reporting groups that did not make AYP by 

status or safe-harbor. The other states, in contrast, only used growth results for non-proficient 

students within reporting groups that did not make AYP by growth or safe-harbor. In those 

states, the non-proficient students without matched data would continue to be counted as nonproficient and would not be subtracted from the denominator for their respective reporting 

group’s AYP calculations. 

In order to gain some information on how the results of growth models like the ones piloted in 

Delaware, Florida, Ohio, and Tennessee might be affected by match rates, matched and nonmatched students were compared in seven of the states (Iowa and Tennessee did not provide 

sufficient information to calculate match rates). Exhibit 56 presents the match rates and average 

test score differences among matched and non-matched students in the states that provided scalescore data to the evaluation project. The match rates are on the number of students who have the 

two years (2006–07 and 2007–08 school years) of test scores needed to calculate growth. The 

number of students with two years of data was divided by the number of students in the indicated 

grade levels who had either or both a reading and a mathematics test score in the 2007–08 school 
Evaluation of the Growth Model Pilot Project 92

year. The seven states generally had match rates of 91 percent or higher based on the supplied 

student data (Exhibit 56). 

Exhibit 56

Two-Year Match Rates for Students and Differences Between Matched and Unmatched 

Grade-Standardized Student Scores for Reading and Math, by State, 2007–08

Pilot States Grades a

Number of 

Students w/ 

Test Scores 

from 2007–08

Percent with 

Test Scores 

from 2006–07

Difference 

(SDs) In 

Reading

Difference 

(SDs) in Math

Alaska 4–8 46,692 92% 0.08 0.14

Arizona 4–7 286,455 95% 0.39 0.37

Arkansas 5–7 106,133 91% 0.26 0.27

Delaware 4–8 43,332 93% 0.19 0.19

Florida 4–8 965,325 91% 0.37 0.40

North Carolina 3–8 663,504 93% 0.48 0.58

Ohio 4–8 627,208 93% 1.65 1.56

Exhibit reads: Of the 46,692 students in grades four through eight in Alaska who had either a 

reading or mathematics test score in 2007–08, 92 percent had matched scores. The mean 

achievement difference in scale scores between those who were matched and those who were 

unmatched was 0.08 standard deviations in reading, and 0.14 standard deviations in mathematics.

a

Source: The Alaska, Arizona, Arkansas, Delaware, Florida, North Carolina, and Ohio state departments of 

education

Iowa and Tennessee did not provide sufficient information to calculate match rates.

In these seven states, reading and mathematics scale scores were available and the mean scores 

of matched and unmatched students could be compared. Unmatched students had substantially 

lower average test scores than matched students. Taken together, these results indicate that while 

the match rates were high, the matched population may have significantly higher levels of 

achievement than the unmatched group and thus be more likely to be proficient. The 

implications are less clear for the proportion of students identified under the growth models to be 

on-track to proficiency. As seen in the section on the effects of different types of models, lowerperforming students are more likely to be classified as on-track under trajectory and transition 

matrix models than under projection models. To that extent, reliance on growth-only data for 

AYP determinations may increase the likelihood a school will reach its AMO under a projection 

model but may not have that effect under the other models. While the match rates in the pilot 

states that provided the information were generally very high, the potential for a few exclusions 

to affect AYP results suggests that further study is needed to determine the validity of using nonmatched status results rather than excluding them from the growth-only applications. In any 

case, these points underscore the importance of maintaining high match rates if growth-only 

outcomes are applied to accountability decisions.

Conclusions

This chapter addressed a number of hypothetical questions about the pilot growth models. The 

first question addressed was the extent to which the results of using only the growth models’ on-
Evaluation of the Growth Model Pilot Project 93

track-to-proficiency results would diverge from the status model results among the schools that 

made AYP by the status model in 2007–08. This analysis identified the schools currently 

performing sufficiently well to make AYP by status criteria but, according to the state’s growth 

model, not making sufficient annual gains to reach the AMO by growth-only criteria. These 

schools would then be expected not to continue to make AYP if their pattern of low growth 

continued. For all nine states combined, 62 percent of the schools that made AYP by status also 

met their reading and mathematics AMOs using just the growth criteria. The states varied 

considerably, ranging from only 46 percent in Arizona and 47 percent in Arkansas and North 

Carolina to 75 percent or more in Ohio, Delaware, and Tennessee. 

The percentages of schools that made AYP by safe-harbor and that also met or exceeded their 

AMO under the growth-only criteria were much lower (28 percent overall). The percentages did 

not exceed 30 percent in any states except Arkansas (64 percent) and Ohio (45 percent).

The largest share of this chapter consisted of an analysis of the effects of different types of 

growth models on student on-track and school AYP determinations. The projection model 

functions in stark contrast with transition matrix and trajectory models. When a projection 

model classifies a non-proficient student as on-track, the probability that a transition matrix or 

trajectory model agrees is near zero, and vice versa. Agreement rates for non-proficient students 

are around 60 percent due to agreement about students not on track.

For status-plus-growth models, projection models will have the least impact, affecting only 

10–20 percent of eligible (non-proficient) students while transition matrix and trajectory models 

affect over 20 percent. States with higher cut scores will have heightened differences between 

the model types and a lower proportional impact of projection models on eligible students. In 

contrast, for growth-only models, projection models will classify the greatest numbers of 

students as on-track.

The models do not yield large differences in the percentages of schools making AYP when nonproficient students who are on-track to proficiency according to each type of model are added to 

the numbers of students meeting or exceeding the proficiency cut point. The models differ much 

more when a growth-only calculation is made first, followed by growth-plus-status calculations. 

Under the growth-only simulation using realistic cut scores and standard AMOs, very few 

schools would make AYP with a trajectory or transition matrix model while most schools would 

make AYP with a projection model.

The student-level results from the generic model comparisons were used to compare the models 

in terms of their predictive accuracy of on-track students actually attaining proficiency. 

Projection models have the highest correct classification rates for future proficiency: over 80

percent. These rates are 5 to 20 percentage points higher than trajectory and transition matrix 

models, depending on the grade level and proximity to the growth model time horizon.

The transition matrix model acts as a coarse, categorical approximation of the trajectory model, 

with agreement rates over 90 percent. The overlap is greatest when time horizons for the 

trajectory model are long or when the number of categories for the transition matrix model is 

large. 
Evaluation of the Growth Model Pilot Project 94

An alternative standard of annual growth to the ESEA-mandated growth-to-proficiency models is 

the difference between the proficiency cut scores in successive grade levels. While not 

acceptable under current regulations for determining school AYP, students gaining that amount 

or more on a vertically scaled assessment would be considered to make “adequate yearly 

growth” regardless of whether they are proficient or on-track to become proficient. Adding the 

non-proficient students who met this alternative growth standard to the pool of proficient 

students would increase the overall rates of students who are arguably performing adequately 

(i.e., proficient or making reasonable progress over the past year).

The final aspect of growth models examined here was match rates. In this chapter we estimated 

the extent to which the pilot states had reading and mathematics test scores from both 2007–08

and 2006–07 available, and whether the 2007–08 test scores differed for the students with and 

without the 2006–07 scores. The results indicate that while the match rates were high (91

percent or greater in the seven states providing the necessary data), the matched population may 

have significantly higher levels of achievement than the unmatched group and thus be more 

likely to be proficient. 
Evaluation of the Growth Model Pilot Project 95

Conclusions

This report was designed to answer two main questions about the implementation of the Growth 

Model Pilot Project (GMPP) under the Elementary and Secondary Education Act, as amended by 

the No Child Left Behind Act of 2001. The two questions and brief summaries of the study 

results based on analyses of data provided by nine pilot grantee states from 2006–07 and 

2007–08 that are the subject of this evaluation (Alaska, Arizona, Arkansas, Delaware, Florida, 

Iowa, North Carolina, Ohio (for 2007–08 only), and Tennessee) are as follows:

How have states in the pilot project implemented growth models?

While the models approved for the nine pilot grantee states that are the subject of this evaluation 

differ from one another in a number of important ways, all use state-specific assessment data to 

measure student progress and proficiency, and the method of incorporating growth outcomes in 

adequate yearly progress (AYP) determinations was generally the same. Eight of the nine pilot 

states applied growth criteria only after schools have failed to make AYP under the status and 

safe-harbor provisions, rather than determining AYP solely on the basis of student 

improvement—a “status plus growth” model. Delaware, in contrast, applied growth criteria first, 

and then applied the status model and safe-harbor provisions, respectively, to schools that did not 

make AYP under the growth model.

When a school is designated as “making AYP under growth,” this means that use of the growth 

model changed the designation for one or more targeted ESEA subgroups. Within the affected 

subgroup(s) for a given school, the growth criterion is usually applied only to the students who 

did not achieve at or above the proficiency level. Simply stated, “making AYP under growth,” 

as defined by the GMPP, does not mean that all students are on-track to proficiency, and it can 

even mean only one non-proficient student is on-track if a sufficient number of others in the 

subgroup are proficient. Eight of the nine pilot states used the status-plus-growth model (Alaska, 

Arizona, Arkansas, Florida, Iowa, North Carolina, Ohio, and Tennessee). Delaware used a 

growth-plus-status model for determining AYP. 

The models approved for the pilot study vary in how they (1) establish growth expectations for 

students, and (2) determine whether individual students are “on-track” to reach proficiency in the 

allotted time frame. We have identified three basic types of growth models being used in the 

GMPP: the transition matrix model (which evaluates student progress from year to year in terms 

of a relatively small set of discrete performance levels), the trajectory model (which uses the gap 

between a baseline test score and a performance standard several years out to calculate the 

amount of growth required to become proficient), and the projection model (which uses current 

and past test scores to statistically predict performance several years ahead, using new test scores 

to update projections of student performance). 

States using a given type of model varied greatly in the extent to which schools were identified 

as making AYP by growth within the GMPP framework. Despite the lack of consistency from 
Evaluation of the Growth Model Pilot Project 96

state to state, the choice of model type can be consequential, as demonstrated in Chapter IV with 

the side-by-side comparisons of generic versions of the three types of models using a common 

dataset. However, the variability in state-by-state AYP results is consistent with the latitude that 

states had in how they defined proficiency levels and implemented (or not) the various 

provisions within ESEA for recognizing AYP under the status model including confidence 

intervals and multiyear averaging. Those provisions (and safe-harbor) are generally designed to 

reduce the chance of incorrectly identifying schools as not making AYP, and the growth models 

were primarily adopted to further reduce that chance. But if the status provisions were already

capturing most of the schools that might have been identified as making AYP by growth, then 

the numbers of schools identified as growth schools by the GMPP would be correspondingly 

reduced.

How did each pilot state’s growth model affect the number and kinds of schools that 

make AYP?

The nine pilot states in 2007–08 that are the main subject of this evaluation provided both 

school-level (EDFacts) and the student-level data to the evaluation project and these data were 

used to address five sub-questions. 

How many schools made AYP under the growth model that would not have made it under the 

ESEA status model?

The designs of the growth models in these states included only those students who (a) did not 

reach proficiency levels in reading or language arts and mathematics, and (b) were members 

of ESEA reporting groups that did not reach their Annual Measurable Objectives (AMOs) or 

obtain AYP via safe-harbor provisions. The pilot models simply added growth criteria to the 

traditional status plus safe-harbor model for determining AYP; thus growth criteria could 

only increase the number of schools making AYP. The number of schools identified as 

making AYP by growth under the GMPP in 2007–08 ranged from 2 percent or fewer of all 

schools in Alaska, Arizona, Iowa, North Carolina, and Tennessee, to 3 percent in Delaware, 5 

percent in Florida, 6 percent in Arkansas, and 34 percent of all schools in Ohio. The schools 

making AYP uniquely by growth represented percentage increases in the numbers of schools 

making AYP that ranged from highs of 102 percent in Ohio, 24 percent in Florida, and 10

percent in Arkansas, to lows of 4 percent or fewer in Alaska, Arizona, Delaware, Iowa, North 

Carolina, and Tennessee. Expressed as percentages of the schools that did not make AYP 

under status or safe-harbor, the impact of the GMPP on identifying additional schools as 

making AYP ranged as high as 50 percent in Ohio, 13 percent in Arkansas, and 10 percent in 

Tennessee.

Are AYP outcomes under the growth models related to school demographics and 

organizational characteristics?

Schools enrolling higher proportions of low-income and minority students were more likely 

to make AYP under growth in the status-plus-growth framework than were schools enrolling 

higher proportions of more affluent and nonminority students. 
Evaluation of the Growth Model Pilot Project 97

How many schools that made AYP under the ESEA status model would also have made it if 

the growth criteria were used exclusively for assessing whether schools met their AMOs?

This question was addressed by drawing on the student data provided by the pilot states to 

calculate whether each school met its AMOs for reading and mathematics using the on-track

to proficiency indicators for each student. For all nine states combined, 67 percent of the 

schools that made AYP by status also met their reading and mathematics AMOs using just 

the growth criteria. Results varied widely among the states: only 46 percent of the Arizona 

schools that made AYP by status also met their AMO using growth criteria alone, but almost 

all of the Ohio (99 percent) and 82 percent of the Tennessee schools that made AYP by status 

also met their AMO with the growth criteria. 

What is the relationship between AYP status under the growth model and ESEA safe-harbor

provisions?

Eight of the nine pilot states had at least one school classified in EDFacts as having made 

AYP by safe-harbor provisions. The percentages of schools that made AYP by safe-harbor

and that also met or exceeded their AMO under the growth-only criteria were much lower 

(38 percent overall) than schools that made both status and growth-only and were 

inconsistent across the eight states with safe-harbor schools, ranging from 4 percent in 

Alaska to 90 percent in Ohio.

How do the main types of models compare in terms of the number of students they identify as 

on-track and the number of schools that meet their AMO by growth criteria?

The projection model functions in stark contrast with transition matrix and trajectory models. 

When a projection model classifies a non-proficient student as on-track, the probability that a 

transition matrix or trajectory model agrees is near zero, and vice versa. Overall agreement 

rates for non-proficient students are around 60 percent due to agreement about students not 

on track.

The transition matrix model acts as a coarse, categorical approximation of the trajectory 

model, with agreement rates over 90 percent. The overlap is greatest when time horizons for 

the trajectory model are long or when the number of categories for the transition matrix 

model is large.

For growth-plus-status models applied to a common dataset, projection models will have the 

least impact, affecting only 10–20 percent of eligible (non-proficient) students while 

transition matrix and trajectory models affect over 20 percent. States with higher cut scores 

will have heightened differences between the model types and a lower proportional impact of 

projection models on eligible students. In contrast, for growth-only models, projection 

models will classify the greatest numbers of students as on-track.

The models do not yield large differences in the percentages of schools making AYP when 

non-proficient students who are on-track to proficiency according to each type of model are 

added to the numbers of students meeting or exceeding the proficiency cut point. The 

models differ much more when a growth-only calculation is used. Under the growth-only 

simulation using realistic proficiency cut scores and standard AMOs, very few schools would 
Evaluation of the Growth Model Pilot Project 98

make AYP with a trajectory or transition matrix model while nearly 30 percent of schools 

would make AYP with a projection model.

How do the main types of models compare in terms of their predictive accuracy of on-track 

students actually attaining proficiency?

When models are applied to a common dataset, projection models were shown to have the 

highest correct classification rates for future proficiency: over 80 percent. These rates are 5 

to 20 percentage points higher than trajectory and transition matrix models, depending on the 

grade level and proximity to the growth model time limit.

How would AYP results change if the growth standards were not tied to attainment of

proficiency within the growth models’ time horizons?

Designed to assess progress toward proficiency within a four-year-or-less time frame, the 

pilot growth models can require very large annual growth increments for students who start 

at low levels of achievement. If ESEA regulations were changed, other kinds of growth 

models than the currently required growth-to-proficiency models could be used for AYP 

determinations. One alternative standard of annual growth that could be used with vertical 

test score scales is the difference between the proficiency cut scores in successive grade 

levels. Students gaining that amount or more would be considered to make “adequate yearly 

growth” regardless of whether they are proficient or on-track to become proficient. A 

simulation shows that the percentages of students meeting that alternative standard of growth 

are lower than the percentage of proficient students in both reading and mathematics, but that 

adding the non-proficient students who met the growth standard to the pool of proficient 

students would increase the overall rates of students who are arguably performing adequately 

(i.e., proficient or making reasonable progress over the past year).

To what extent are longitudinal student data required by the growth models available?

We estimated the extent to which the pilot states had reading and mathematics test scores 

from both 2007–08 and 2006–07 available, and whether the 2007–08 test scores differed for 

the students with and without the 2006–07 scores. While the match rates were high (90 

percent or greater in all states), the matched population may have significantly higher levels 

of achievement than the unmatched group and thus be more likely to be proficient or on-track

to proficiency. To that extent, reliance on growth data with their requirement for matched 

longitudinal records for AYP determinations may increase the likelihood a school will reach 

its AMO.

Implications for Future Policy

Implications for future policy relate to reporting of growth model results, selection of growth 

models, and use of growth models not tied to proficiency standards. 

Reporting of growth model results

As used for school AYP purposes in all states except Delaware, the growth information for 

students was used only for reporting groups that did not make AYP under status criteria or safeharbor provisions. Consequently, it was generally not possible to determine from the official 

EDFacts reports the extent to which a school designated as making AYP by growth was actually 
Evaluation of the Growth Model Pilot Project 99

realizing growth among its students. Delaware applied growth criteria before status and safeharbor criteria and all schools identified as making AYP by growth actually met their AMOs on 

the basis of Delaware’s growth model. An implication for future policy is that other states could 

clarify their schools’ progress by applying growth criteria before status and safe-harbor. Two 

general ways of doing that can be identified and each has advantages and disadvantages. Both of 

these options are methods for reporting schools’ results related to AYP determinations and do 

not suggest changes that would actually affect whether schools are determined to make AYP. As 

described here, neither of the methods would affect a school’s AYP status. However, the second 

option is recommended because it fully establishes and reports both the growth and the status 

information.

Reporting option #1: Applying growth criteria before status and safe-harbor criteria to all 

schools’ AYP determinations

A different possible ordering would involve first using the growth results to assess whether the 

reporting group reached the AMO, then, if not, using the union of growth and status results to 

estimate the percentage of students on-track or proficient. Safe-harbor might only then apply to 

reporting groups that did not reach the AMO through either of these calculations.

The main advantage of applying growth before status and safe-harbor is that it would identify 

schools that are realizing adequate progress toward universal proficiency. This would clearly 

distinguish those schools from schools making AYP under status criteria but not realizing growth

sufficient to continue meeting their AMOs. The latter are probably not a large number but 

identifying such schools would serve as an early warning mechanism of possible problems. The 

exploratory analyses in this report also indicate that applying growth criteria before safe-harbor

could usefully reclassify most of the current safe-harbor schools as making AYP by growth and 

would clearly identify the minority that are not on-track to proficiency and thus headed for 

sanctions in the near future. 

One disadvantage is that schools that are making AYP both by status and growth (arguably the 

strongest schools) would not be uniquely identified. Another possible disadvantage of applying 

growth first follows from the fact that the AYP-by-growth determination would be based on 

students with at least two consecutive years of test data. Matched students are likely to have 

higher test scores and thus make it more likely that their subgroups and school reach the AMO. 

While the match rates in the pilot states that provided the information were generally very high, 

the potential of a few exclusions to affect AYP results suggests that states consider using nonmatched students’ status results rather than excluding them from otherwise growth-only 

applications.

Reporting option #2: Applying both growth and status criteria to all schools’ AYP 

determinations

The GMPP required states to compile student-level data on both status and growth criteria, and 

these data could be used to classify each school simultaneously on both criteria for AYPreporting purposes. That is, each school could be classified as making AYP by both growth and 
Evaluation of the Growth Model Pilot Project 100

status, by growth-only, by status-only, by a mix of status and growth, or by safe-harbor only. 

This would have the advantage of uniquely identifying different sets of schools (those making 

AYP in terms of both growth and status). A possible hierarchy for reporting purposes would be:

• Met AMO by growth and by status (best)

• Met AMO by growth but not status

• Met AMO by status but not growth

• Met AMO by combining status and growth results, so that status results are used for 

students not on-track to proficiency

• Did not make AMO (worst)

A disadvantage is that some states (e.g., Delaware and Iowa) classified all students who were 

proficient using the status criterion as also being on-track to proficiency on the growth criterion, 

regardless of their prior year scores. This would have the effect of increasing the number of 

“met by growth and status” schools and would prevent comparisons with states that measured 

growth for proficient and non-proficient students. However, the transition matrix models used 

by those states could be extended to include categories above proficiency and thus allow for 

growth-only inferences for proficient students.

Selecting among alternative growth models in the current policy context

This study has also shown that the types of growth models states select for federal accountability 

purposes are consequential and raise some potentially difficult theoretical questions for 

policymakers. Projection models are likely to be more accurate than transition matrix or 

trajectory models in terms of predicting students’ future attainment of proficiency targets but the 

simpler trajectory and transition matrix models may provide clearer guidance to schools, 

teachers, students, and their parents. 

The projection model is explicitly designed to provide probabilistic predictions whereas the other 

models do not. As a probabilistic estimator, the projection model carries a measure of 

uncertainty for each student’s predicted score. An important issue illustrated by the Ohio model 

is whether to adjust a student’s predicted score for the uncertainty and, if so, to what extent. The 

adjustment used by Ohio (adding two standard error units to each student’s predicted score) was 

selected in order to make it highly unlikely that a student who actually was on-track was 

misclassified as not on-track. However, the adjustment also had the effect of classifying a much 

higher percentage of non-proficient students as on-track than the other pilot states. Careful 

consideration of the trade-off between false-negatives and false-positives is needed when 

adjustments for uncertainty are made to statistically derived growth model results such as those 

from projection models.
Evaluation of the Growth Model Pilot Project 101

Alternatives to proficiency-based growth targets

The growth targets identified by the growth models are tied to the ESEA goal of universal 

proficiency by 2014. This means that substantial individual student performance improvements 

that do not reach the growth models’ proficiency targets (generally within three or four years, or 

by grades 8 or 9) or subgroup- and school-aggregate student performance improvements that do 

not reach the 2014-driven proficiency targets (AMOs) are not recognized by the GMPP growth 

models. However, while not permitted under current ESEA regulations, growth models could be 

adapted to other targets with the result that more students and schools would be identified as 

making adequate progress than is currently the case. Such other targets generally involve use of 

a more finely graduated set of performance outcomes than proficiency or on-track to proficiency 

and moving away from the goal of universal proficiency by 2014.
Evaluation of the Growth Model Pilot Project 102
Evaluation of the Growth Model Pilot Project 103

References

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Measurement: Issues and Practice, 28(4), 42–51.

Booher-Jennings , J. (2005). “Below the Bubble: ‘Educational Triage’ and the Texas 

Accountability System.” American Educational Research Journal, 42: 231–268.

Chudowsky, N., Chudowsky, V., and Kober, N. (2009). Is the Emphasis on “Proficiency”

Shortchanging Higher- and Lower-Achieving Students? Center for Education Policy, 

Washington, D.C. Retrieved July 2009 from http://www.cepdc.org/index.cfm?fuseaction=document_ext.showDocumentByID&nodeID=1&Documen

tID=280

Council of Chief State School Officers (CCSSO) (2008). Implementer’s Guide to Growth. 

Washington, D.C. Retrieved May 2008 from 

http://www.ccsso.org/content/pdfs/IGG%20Final%20AP.pdf. 

Ho, A. D., Lewis, D. M., and Farris, J. L. M. (2009). The dependence of growth-model results on 

proficiency cut scores. Educational Measurement: Issues and Practice, 28(4), 15–26.

Neal, D., and Schantenback, D. (August 2007). National Bureau of Economic Research, 

Working Paper 13239. Left Behind By Design: Proficiency Counts and Test-Based 

Accountability. Cambridge, MA. Retrieved September 2008 from 

http://www.nber.org/papers/w13293.pdf.

U.S. Department of Education, Office of Elementary and Secondary Education (Jan. 15, 2009). 

Evaluation of the 2005–06 Growth Model Pilot Program. Washington, D.C. Retrieved 

Feb. 2009 from http://www.ed.gov/admins/lead/account/growthmodel/gmeval0109.doc.

U.S. Department of Education, Office of Elementary and Secondary Education (July 2007). No 

Child Left Behind Fact Sheet: “Growth Models: Ensuring Grade-Level Proficiency for 

All Students by 2014.” Washington, D.C. Retrieved May 2008 from 

http://www.ed.gov/admins/lead/account/growthmodel/proficiency.pdf.

U.S. Department of Education (Jan. 25, 2006). “Peer Review Guidance for the NCLB Growth 

Model Pilot Applications.” Washington, D.C. Retrieved May 2008 from 

http://www.ed.gov/policy/elsec/guid/growthmodelguidance.pdf.

U.S. Department of Education, Office of the Secretary, Office of Public Affairs (2003). No Child 

Left Behind: A Parent’s Guide. Washington, D.C. Retrieved May 2008 from 

http://www.ed.gov/parents/academic/involve/nclbguide/parentsguide.pdf.
Evaluation of the Growth Model Pilot Project 104

U.S. Department of Education (2008). “U.S. Secretary of Education Margaret Spellings 

Approves Additional Growth Model Pilots for 2007–2008 School Year,” press release, 

June 10, 2008. Retrieved June 2008 from 

http://www.ed.gov/news/pressreleases/2008/06/06102008.html. 

U.S. Department of Education, Office of Communications and Outreach (2009). “U.S. Secretary 

of Education Margaret Spellings Approves Additional Growth Model Pilots for 2008–

2009 School Year”, press release, Jan. 8, 2009. Retrieved January 2009 from 

http://www.ed.gov/news/pressreleases/2009/01/01082009a.html.

U.S. Department of Education, Office of Communications and Outreach (2006). “Secretary 

Spellings Approves Additional Growth Model Pilots for 2006–2007,” press release, Nov.

9, 2006. Retrieved June 2008 from 

http://www.ed.gov/news/pressreleases/2006/11/11092006a.html. 

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Spellings Announces Growth Model Pilot, Addresses Chief State School Officers’

Annual Policy Forum in Richmond,” press release, Nov. 18, 2005. Retrieved May 2008 

from http://www.ed.gov/news/pressreleases/2005/11/11182005.html.

U.S. Department of Education, Office of Planning, Evaluation and Policy Development, Policy 

and Program Studies Service (2007). State and Local Implementation of the No Child Left 

Behind Act, Volume III—Accountability Under NCLB: Interim Report, by K. LeFloch, F. 

Martinez, J. O’Day, B. Stecher, J. Taylor, and A. Cook. Washington, D.C.: Author.
Evaluation of the Growth Model Pilot Project 105

Appendix A:

Comparison of GMPP Growth Models with 

State Accountability Systems

Of the nine states approved for the GMPP for the 2007–08 school year, six (Arizona, Arkansas, 

Florida, North Carolina, Ohio, and Tennessee) already had implemented formal measures of 

student growth for state accountability purposes before their GMPP models were approved and 

another (Alaska) implemented a state growth model concurrently with starting the in GMPP. 

Iowa included growth as an optional measure. In comparison, a 2007 survey by the Council of 

Chief State School Officer’s (CCSSO) Accountability Systems and Reporting State 

Collaborative on Assessment and Student Standards (ASR-SCASS) found that only six of the 23 

responding states had fully operational non-AYP growth models and fewer than half calculated 

any form of growth at all.39 This suggests that states with growth modeling experience were 

better positioned to apply for and receive approval to participate in the federal program. In fact, 

the two states first approved for GMPP had the most prior experience using student progress to 

grade schools. Exhibit A.1 provides brief descriptions of each state’s pre-GMPP growth model.

 39 See http://edmeasure.com/ASR/doku.php?id=asr:survey_results (retrieved May 2008). A GAO survey in March 

2006 found that 26 of 50 states were using growth models, of which seven measured individual student growth 

(GAO–06–948T). Three of those states are in the GMPP, and our review of documentation suggests that three other 

GMPP states were using (or about to use) student growth information for state accountability purposes.
Evaluation of the Growth Model Pilot Project 106

Exhibit A.1

Overview of Growth Components of State Accountability Systems

State

Non-AYP 

Growth 

Model

Year 

Implemented

Grades and 

Subjects 

Covered Growth Standards Applied School Progress Measured Other Features

Alaska School 

Performance 

Incentive 

Program

2007–08 3–10

Reading/Writing/ 

Math

Growth is measured using 

seven categories of 

performance.

Points are awarded for 

student movement among 

performance categories and 

are averaged to create a 

school index score.

Four performance levels are 

used to reward schools 

according to their growth index 

scores.

Arizona Measure of 

Academic 

Progress 

(MAP)

1999–2000, 

revised 2005–

06

4–8

Reading/Math

Growth is measured using 

annual differences in scale 

scores and is averaged to 

calculate school growth.

Points are awarded for 

meeting expected growth (by 

quartile) and are added to 

other measures of school 

performance (AZ LEARNS).

Expected growth is calculated 

using a value-added model 

adjusted to ensure proficiency 

by seventh grade and 

controlling for mobility or 

ceiling effects.

Arkansas Act 35 Annual 

School 

Ratings 

System

2003–04, 

growth added 

in 2008–09

3–11

Reading/Writing/ 

Math

Student growth is based on 

changes in eight 

performance levels across 

adjacent years. 

Value points are added if a 

student moves up a 

performance category and 

are lost if a student moves 

down a category.

Schools are evaluated using 

five performance levels for 

both “improvement” (Category 

One) and “status” (Category 

Two).

Delaware n/a n/a n/a n/a n/a n/a

Florida Grading 

Florida Public 

Schools

2001–02 3–10

Reading/Math/ 

Science/Writing

Growth is measured by 

movement among five 

performance levels.

Grades schools by awarding 

points for the number of 

students meeting proficiency 

standards plus those making 

annual learning gains.

Points also are awarded for 

students who maintain high 

performance, for the percent of 

students in the lowest quartile 

making gains, and if half or 

more of such students make 

gains.

Iowa Optional n/a K–12

Reading/Math/

Social Studies/

Science

Growth for student profiles is 

measured using changes in 

ITBS and ITED scores.

n/a n/a

Exhibit A.1 continues next page
Evaluation of the Growth Model Pilot Project 107

Exhibit A.1

Overview of Growth Components of State Accountability Systems

continued from previous page

State

Non-AYP 

Growth 

Model

Year 

Implemented

Grades and 

Subjects 

Covered Growth Standards Applied

School Progress 

Measured Other Features

North 

Carolina

ABCs Growth 

Model

1996–97, 

revised 2005–

06

K–8, HS

Reading/Math/ 

Science

Growth is the average of 

standardized scores from 

two previous years adjusted 

for the tendency of students 

with extremely high or low 

initial score to score closer 

to the mean on the next 

assessment (“regression to 

the mean”) and is expected 

to be “0” or above, indicating 

at least one year’s worth of 

gain.

A mean growth score is 

calculated for the school, 

though schools can also 

meet expected growth if 60 

percent or more of students 

test at grade-level 

proficiency.

Schools get credit for “high 

growth” if students who are 

stable or growing outnumber 

those declining by at least 

three to two [or a ratio of 3:2].

Ohio Value-Added 

Assessment

2006–07 4–8, plans for 

K–10

Reading/Math

Growth is measured as 

mean normal curve 

equivalent (NCE) gains and 

is categorized as meeting or 

as being a standard error 

“above” or “below” the 

“expected” gain.

LRC designations can be 

raised if schools are above 

expected growth for at least 

two years and lowered if 

below expectations for at 

least three years.

Value-added is based on the 

2006–07 distribution of test 

scores and will be used for 

accountability beginning in 

2007–08.

Tennessee Tennessee 

Value-Added 

Assessment 

System 

(TVAAS)

1992–93 3–8

Reading/Math/ 

Language/ 

Science/Social 

Science

Growth in TCAP results is 

measured as mean normal 

curve equivalent (NCE)

gains relative to a state 

growth standard (1998 Terra 

Nova) or a state three-year 

average.

School reports include 

average student gains by 

grade and whether 

achievement level meet 

growth standards or are 

within one, two, or more 

standard deviations.

Specific school effects also 

are estimated using up to five 

years of student data to 

predict scores for the average 

student in a given school; 

teacher effect data is 

restricted access.
Evaluation of the Growth Model Pilot Project 108

However, these state accountability growth models had to be modified in significant ways for 

use in GMPP. One basic difference between models approved under ESEA and those used for 

state accountability purposes is the definition of expected growth. The seven core principles 

require that the ESEA growth models place limits on the amount of time students have to reach 

proficiency, usually by the last grade tested as applied, resulting in growth expectations that put a 

student “on-track” to meet grade-level standards in a few years. Most state growth models do 

not require students to be on-track to proficiency but, for example, simply expect students to 

exhibit a year’s worth of growth per year of instruction. The principle of universal proficiency 

by 2014 also translates into rising AMOs, a requirement that is absent from state accountability 

systems. The goal of states in most cases is to determine whether individual schools and the 

system as a whole are improving rather than to see if a given student will be proficient in a few 

years or if a school will have all students proficient by 2014. 

Another major difference is that growth components of state accountability systems typically 

calculate an average for the entire school instead of basing school performance on the results of 

each subgroup separately and may average over subjects as well. This means that a school can 

still make adequate growth for state accountability purposes even if one or more subgroups fails 

to grow or if the school exhibits strong growth in one subject but not another. Of course, this 

does not mean that growth results for subgroups and subjects are not reported, and states often 

measure growth for more than reading and math. The goal is to create a composite growth score 

that can then be combined with other measures of school performance (including AYP) both for 

reporting purposes, such as a “grade” on school report cards, and for rewarding schools that meet 

or exceed state standards. 

The result of these differences is that more schools tend to make adequate growth for state 

accountability purposes than would make AYP under the GMPP. For example, 779 of the 1,298 

(or 60 percent) of the North Carolina schools that failed to make AYP in 2006–07 made expected 

growth under the state guidelines. Of these, 187 (or 14 percent) met the state standards for high 

academic growth. The federal growth model used by North Carolina sets the bar higher for both 

expected growth and percent proficient (AMO) than does the growth model used for state 

accountability, suggesting that the universal proficiency requirement is driving most if not all of 

this difference. Florida also drops the proficiency deadline in its grading system, although the 

transition matrix model used sets expectations that may differ from a single year’s worth of 

growth.40

Offsetting this tendency of the 2014 deadline to lower the number of schools making adequate 

growth is that state accountability systems do not give schools credit if enough students grow to 

push a subgroup or school over the proficiency threshold as is done in the status-plus-growth 

 The school itself can make adequate growth if half or more of low-achievers meet 

expected growth, which is a different standard than the AMO used for AYP purposes. Thus 

about 84 percent of Florida schools made adequate growth in both reading and math for 

2006–07, compared to the 34 percent of schools that made AYP under the GMPP. 

 40 Note that students who score in the two lowest FCAT achievement levels still can make expected growth if they 

make a year’s worth of growth as determined by FCAT developmental scores.
Evaluation of the Growth Model Pilot Project 109

models implemented under the GMPP.41 This means that schools classified as making AYP by 

growth may not be showing much growth for the student body as a whole. Thus of the 12 North 

Carolina schools that made AYP by growth in 2006–07, only three met the state’s high growth 

standard and one did not make the lower standard of expected growth. Similarly, 10 of the 19 

Tennessee schools that made AYP by growth in 2006–07 grew at a rate below the state threeyear average for math, while 14 schools grew at a below average rate for reading. One school 

even experienced negative growth for both reading and math. Growth models are only one 

component of federal accountability under the GMPP and thus do not provide much information 

about whether a school is growing as currently reported. However, results from growth 

measures used for state accountability purposes suggest that many more schools would make 

AYP if the first of the seven core principles of the ESEA project was relaxed.

 41 Note that states often give growth credit to schools for maintaining proficiency, which is similar to a growth-plusstatus model, and may grade schools on proficiency status criteria as well.
Evaluation of the Growth Model Pilot Project 110
Evaluation of the Growth Model Pilot Project 111

Appendix B:

State GMPP Model Summaries

The core of each growth model is the method of determining whether a student is “on-track” to 

being proficient. While states’ methods can be categorized as one of three types of growth 

models (trajectory, transition matrix, or projection), each state’s method has unique features. 

This appendix offers additional details about the growth models used in Alaska, Arizona, 

Arkansas, Delaware, Florida, North Carolina, and Tennessee. All the relevant detail of Iowa’s 

growth model is included in the main text. Ohio used the same growth model as Tennessee and 

the latter’s summary is sufficient for both states with the important difference that Ohio adjusted 

predicted scores by the standard errors of the predictions (see Ohio summary above).

Alaska

Alaska uses a trajectory model that calculates growth for students in grades 4 through 6, grade 8 

and grade 9, using results of the Alaska Standards Based Assessment Test mathematics and 

language arts. 

Proficiency levels for these tests are scaled so that thresholds are 300 for math and 600 for 

literacy within each grade. The literacy score is cutoff at 600 because it is a combination of 

reading and writing, each of which has a cutoff of 300 points. For this summary we will 

consider a generic cutoff score of 300 for each grade. 

Students are always classified by the traditional status model in grades 3 and 7. In grades 4 

through 6 it is possible to use the growth model. In the Alaska growth model the first year a 

student is below proficient is considered his or her base year, and growth targets for subsequent 

years are calculated by evenly dividing the difference between the base year score and the cutoff 

in grade 7. In grades 8 and 9, the first year a student is below proficient is considered the base 

year, and growth targets for subsequent years are calculated by evenly dividing the difference 

between the base year score and the cutoff in grade 10. 

For example, if a student scores below proficient in some grade q (between grades 3 and 6), 

growth targets for each subsequent grade are a function of his or her base score in grade q, Yq, 

plus the difference between this score and 300 weighted fraction of the years that have elapsed 

between the previous grade and grade 7. Thus, a student in grades 4 through 6 is considered to 

be on-track toward proficiency in grade k if his or her score, Yk

Y Y Y kq q kq q ≥+ − × − − (300 ) (( ) (7 ))

, is greater than or equal to this 

function,

, assuming k > q and Yq

For students who score below proficient in some grade q between grades 7 and 10, growth 

targets for each subsequent grade are a similar function of the student’s base score in grade q, 

< 300.
Evaluation of the Growth Model Pilot Project 112

Yq, plus the difference between this score and 300 weighted fraction of the years that have 

elapsed between the previous grade and grade 10. Thus, a student in grades 8 or 9 is considered 

to be on-track toward proficiency in grade k if his or her score, Yk

Y Y Y kq q kq q ≥+ − × − − (300 ) (( ) (10 ))

, is greater than or equal to this 

function, or, 

, assuming k > q and Yq

These targets, once established by the baseline year, remain the growth targets for subsequent 

years.

< 300.

The illustration in exhibit B.1 below showcases this process. This figure shows the proficiency 

standard for each grade (300) along the solid line connecting points a to i. The shaded polygons 

illustrate the necessary growth between years to be on-track toward proficiency for a set of 

hypothetical students. 

Exhibit B.1

Illustration of Alaska’s Method for Determining Whether a Student Is

On-Track Toward Proficiency

Label a b c d e f g h i

Grade 3 3 4 5 6 7 8 9 10

Score 300 200 226 250 275 200 234 267 300
Evaluation of the Growth Model Pilot Project 113

Student Z is below proficient in the third grade, he scored 200 (point b) instead of the proficiency 

standard of 300. In applying the formulas above, Alaska’s growth model would set a new target 

score of 226 in the fourth grade (point c), 250 in grade 5 (point d), and 275 in grade 6 (point e), 

to be considered on-track for those grades. While that student cannot be used for growth model 

AMO calculations in the third grade, that student will be expected to make those targets 

calculated for subsequent years to be considered as on-track in grades 4 through 6. Similarly, 

student X is below proficient in the seventh grade, he scored 200 (point f) instead of the 

proficiency standard of 300. In applying the formulas above, Alaska’s growth model would set 

new target scores of 234 in the eighth grade (point g), and 264 in grade 9 (point h), to be 

considered on-track for those grades. While that student cannot be used for growth model AMO 

calculations in the seventh grade, that student will be expected to make those targets calculated 

for subsequent years to be considered as on-track in grades 8 and 9. 

Arizona

Arizona uses a trajectory model to set growth targets for students in grades 3 through 8 using 

Arizona’s Instrument to Measure Standards (AIMS) test. The score cutoffs for reading and math 

are presented in Exhibit B.2.

Arizona, like other states using trajectory models, sets growth targets for students who are not 

proficient based on a baseline score. Growth targets are set by determining the necessary 

improvement to reach proficiency and setting evenly spaced improvement targets over a given 

set of years. In Arizona, growth targets are set based on the first non-proficient score. Growth 

targets calculated from a baseline score in K through third-grade require the student to reach 

proficiency by the sixth grade. Growth targets calculated based on a fourth grade score require 

the student to be proficient by seventh grade. Growth targets calculated based on scores in 

grades 5, 6, and 7 require the student to be proficient by the eighth grade. Growth targets are 

reset each year based on student performance and movement in and out of the Arizona public 

school system. However, the specific grade required to reach proficiency does not change when 

targets are reset. 

Exhibit B.2

Arizona Proficiency Standards Test Cutoffs for Grades 3 Through 8

Grade Reading Cutoff Math Cutoff

3 431 420

4 450 448

5 468 476

6 478 496

7 489 517

8 499 537

While each student is assigned growth targets based on actual performance, Arizona is unique 

compared to other growth model pilot states in that schools are evaluated on each student’s 

estimated score instead of students’ actual scores. However, in calculating the proportion of 
Evaluation of the Growth Model Pilot Project 114

students within a school who are considered to be on-track toward proficiency, students are 

classified based on the lower bound of their predicted scores. These predicted scores are 

generated from a statewide statistical model which fit current year scores to scores from the 

previous years. 

Arizona estimates a regression model predicting the current year scores of all students by their 

previous year’s score. Let Yit

Yit = α + βYit−1 + ei

be the score from student i in grade k in school j at the current 

year t, . From this model, Arizona predicts each student’s current score,

1

* Y Y it = + α β it− . There is a statewide standard error, *Y t

SE , associated with each year’s 

predicted score, which typically equals about 4 or 5 scale score points. The score used to 

evaluates a student’s growth, Yit

* , is then a lower bound of a confidence interval estimated around 

the predicted score, or * * 1.96 Lower

it it Y t

Y Y SE   =− ×    . As a result, each student is evaluated not 

on his or her actual score, but on this lower bound of an estimated predicted score. 

The illustration in exhibit B.3 below showcases this process. This figure shows the proficiency 

standards for grades 4, 5, 6, 7, and 8 along the solid line connecting points a, c, i, l, and m. The 

shaded polygons illustrate the necessary growth between years to be on-track toward proficiency 

for a of hypothetical student. 

Student Z is below proficient in the fourth grade, he scored 446 (point b) instead of the 

proficiency standard of 450 (point a). Arizona’s growth model would set a new target score of 

460 in the fifth grade (point e) to be considered on-track. Student Z then scored exactly 454 in 

the fifth grade (point f), which is below the necessary growth target. However, the student 

predicted score is 465 (point d), which is above the growth target. Yet, the school will be unable 

to count that student as on-track because the lower bound of the predicted confidence interval is 

457 (point d*), still better than the actual score, but also still below the growth target. The 

student actual fifth grade score is then used to estimate a new growth target of 469 in the sixth 

grade (point j). 

Student Z then did well in the sixth grade, scoring 484 (point g), well above the normal cutoff of 

478 (point i). However, that student’s predicted score is 480 (point h) with a lower bound 472 

(point h*). Because the lower bound is still greater than the growth target, that student can then 

be counted as on-track. Under a growth only model, that student’s growth target for seventh 

grade would be higher than the normal cutoff (point k instead of point l) because that student’s 

actual sixth-grade score was greater than the normal cutoff as well. 
Evaluation of the Growth Model Pilot Project 115

Exhibit B.3

Illustration of Arizona’s Method for Determining Whether a Student Is 

On-Track Toward Proficiency

Label a b c e d d* f g h h* i j k l m

Grade 4 4 5 5 5 5 5 6 6 6 6 6 7 7 8

Score 450 446 468 460 465 457 454 484 480 472 478 469 491 489 499

Arkansas

Arkansas uses a trajectory model that calculates growth for students in grades 4 through 7 using 

results of the Arkansas Benchmark Exams for mathematics and literacy, which are administered 

in grades 3 through 8 (plus grade 11 for literacy). 

Proficiency levels for these vertically scaled exams are set for each grade, and growth targets are 

based on the annual exam score increment needed to reach the proficiency standard in eighth 

grade. These proficiency standards are presented in table B.4 below.
Evaluation of the Growth Model Pilot Project 116

Exhibit B.4

Arkansas Benchmark Exam Proficiency Standards for Grades 3 Through 8

Grade Proficiency Standard

3 500

4 559

5 604

6 641

7 673

8 700

If a student does not achieve proficiency in a base year, the growth targets necessary to be 

considered on-track toward proficiency are calculated based on the total increment between a 

student’s base score and the proficiency standard in eighth grade (700). However, instead of 

setting growth targets by evenly dividing the total increment between the base score and 700, the 

increments are proportioned so that the growth increments follow a curve that matches the 

concave nature of the typical proficiency standards. 

This means that, for example, instead of setting the growth increment from third to fourth grade 

to be a quarter (0.25) of the total necessary improvement to reach 700 by eighth grade, the 

growth increment for fourth grade is set to slightly more than a quarter (0.295) of the total 

necessary improvement to reach 700 by eighth grade. Thus, the annual increment that a student 

must attain in order to be classified as on-track to proficiency is calculated using grade-specific 

growth target multipliers. The multiplier for any grade “k” is simply a ratio of the difference 

between the current and previous year’s proficiency standards, Pk and Pk-1

Pk − P ( k −1 ) 700 − P ( k −1 )

, to the difference 

between the proficiency standard for eighth grade and the previous year, or, 

.

These multipliers are presented in Exhibit B.5.

Exhibit B.5

Arkansas Growth Target Multipliers for Grades 3 Through 7

Grade Multiplier

4 0.295

5 0.319

6 0.385

7 0.542

Another method of illustrating a specific growth increment is to use the following formula that 

calculates the minimum score necessary to be considered on-track toward proficiency. For any 

student in grade k, the minimum score necessary to be considered “on-track” is his or her 
Evaluation of the Growth Model Pilot Project 117

previous score, Yk-1,, plus the difference between 700 and his or her previous score, 700-Yk-1, 

times the multipliers outlined above. Thus, if the current score, Yk

YY PP P Y kk kk ≥+ − − × − − −− − 1 11 1 ((( ) (700 700 k )) ( k ))

, is greater than or equal to 

this function, or, 

,

that student is considered to be on-track toward proficiency. From this formula, we see that this 

model resets the growth target every year rather than setting a series of annual targets based on 

the first below-proficient exam score. This formula can also determine minimum scores even for 

students who score above the proficiency threshold. Thus, students can be identified who are not 

improving toward the eighth-grade standard of 700 even though they are currently proficient. 

The illustration in exhibit B.6 below showcases each of these features. This figure shows the 

proficiency standards for each grade along the solid line connecting points a, c, g, k, m and n. 

The shaded polygons illustrate the necessary growth between years to be on-track toward 

proficiency for a set of hypothetical students. 

Student Z is below proficient in the fourth grade, he scored 505 (point d) instead of the 

proficiency standard of 559 (point c). In applying the formulas above, Arkansas’ growth model 

would set a new target score of 567 in the fifth grade (point i) to be considered on-track. If 

student Z scored exactly 567 in the fifth grade, then his growth targets would follow the dotted 

line from point i to point n. In addition, imagine that student Z in fact did better than scoring 567 

in the fifth grade—let us say that he scored 575 (point h). Then, that student’s growth target for 

sixth grade would be reset and instead would be 623 (point l). Note that because student Z did 

better than his growth target in the fifth grade, his new growth target for the sixth grade is 

slightly higher—point l is above the dotted line from point i to point n. 

Another student, student X, scored well above the standard in fourth grade at 600 (point b). By 

applying the same formula, the Arkansas growth model would set a fifth-grade target score of 

632 (point e), which is likewise well above the proficiency standard of 604 (point g). Next, 

imagine that student X scored lower than his target score in fifth grade—let us say he scored 615 

(point f). Student X scored above proficient in fifth grade, but because is score is lower than his 

target score generated from fourth grade, he is not considered to be on-track toward proficiency. 

The Arkansas model does reset scores each year, and so we can see that his target for sixth grade 

is set at 648 (point j), which is closer to the proficiency standard of 641 (point k). Note that his 

sixth-grade target is below the dotted line from point e to point n. This shows that his new target 

score is reflecting that he did not score as high in the fifth grade as he did in the fourth grade. 

Still, he would need to score at least 648 in the sixth grade (point j), which is more than the 

normal sixth-grade proficiency standard of 641 (point k). 
Evaluation of the Growth Model Pilot Project 118

Exhibit B.6

Illustration of Arkansas’ Method for Determining Whether a Student 

Is On-Track Toward Proficiency

Label a b c d e f g h i j k l m n

Grade 3 4 4 4 5 5 5 5 5 6 6 6 7 8

Score 500 600 559 505 632 615 604 575 567 648 641 623 673 700

Delaware

Delaware’s transition matrix model uses a “value table” method for AYP determinations that 

assigns points for students depending on the type and extent of changes between the performance 

levels (see Exhibit 6 in the report). Points in the value table increase with both the magnitude of 

change in level of proficiency, and the overall level of proficiency. As a result, a student scoring 

toward the bottom during his previous grade receives fewer points for moving up one level than 

he would if he moved up several levels. All students who surpass their grade level cutoff receive 

the maximum points regardless of how high they scored or whether their underlying scores 

actually declined from the prior year. From these points assigned to students, schools meet their 

AMO if the average number of points per student meets or exceeds a target number of points set 

by Delaware for particular subjects and grades. 

Delaware’s point system is designed to be an analogue to the AMO system used for status 

models by giving partial weight to students who are below proficient but still making progress. 
Evaluation of the Growth Model Pilot Project 119

In fact, the target number of points that Delaware sets is simply the standard AMO that is 

rescaled to the point system. 

For example, in the 2006–07 school year the status model AMO for reading was 68 percent 

proficient. For that same year, the target number of points that a school’s students needed to 

gain on average was 204. If we divide the target points by the maximum number of points a 

student can receive, we return to the AMO (204/300 = 0.68). To illustrate how a school meeting 

the AMO under the status model would perform under the growth model, imagine two schools 

with 100 students, school Z and school X (see exhibit B.7 below). 

In the first school, school Z, exactly 68 out of 100 students (68 percent) were proficient in 

reading. That school meets the AMO of 68 percent exactly. Let us assume that in this school all 

students who were below proficient did not move up any categories from their previous grade. 

Then, under Delaware’s growth model, that school would receive 

points,

out of a total possible 

points.

The average number of points per student would be

,

which is exactly the target number of points a school needs to make their AMO under the growth 

model, and the number of achieved points, 20,400, is exactly 68 percent (the AMO) of the total 

possible 30,000 points. Thus school Z would make their AMO under both the status model and 

the growth model. 

To illustrate how a school that would not meet its AMO under the status model, but would make 

its AMO under the growth model, consider another school X which also has 100 students, but 

only 30 students are proficient. Under the status model, only 30 percent of the students are 

proficient, which is well below the target AMO of 68 percent. Yet, the school does have many 

students who are improving. Under the growth model, the proficient students account for only

points,

which is only 30 percent of the total possible 30,000 points. Remember, though, school X has 

students who are improving, including 30 students who moved from PL 1A to PL 1B for 

points,
Evaluation of the Growth Model Pilot Project 120

and another 15 students moved from PL 1B to PL 2A for 

points,

and another 15 students moved from PL 1B to PL 2B for 

points,

and 5 students who did not improve, who stayed at PL1A for 

5 × 0 = 0 points.

That school would total 20,500 points out of a possible 30,000 points. That would be 68.3 

percent of the total possible points, greater than the AMO for reading that year. The average 

number of points per student would be 205, which is larger than the 204 points necessary to meet 

the AMO. Under the growth model, school X would meet the AMO.

This means that Delaware’s status model is actually like a simple transition matrix in which all 

students who are proficient are given the maximum number of points, and all student who are 

below proficient are given no points at all. The growth model, then, gives students who improve 

a partial number of points. The table in Exhibit B.7 can be used to estimate the proportion of the 

total points each improvement is worth. Another way to think about Delaware’s growth model is 

that a student who moves from the first level, PL 1A, to the second level, PL 1B, is given half 

(0.50) the weight of a student who is proficient when the percent proficient is calculated. This 

means that in Delaware’s transition matrix system, two students who move from the first to the 

second level equal one student who is proficient. 
Evaluation of the Growth Model Pilot Project 121

Exhibit B.7

Example Comparison of Two Hypothetical Schools in

Delaware to Illustrate Growth Model

Year 1 Year 2

Points Per 

Student

Proportion 

of Total 

Possible 

Points

School Z School X

Number of 

Students 

Per Cell

Points 

Per Cell

Number of 

Students 

Per Cell

Points 

Per Cell

PL 1A PL 1A 0 0.00 0 0 5 0

PL 1A PL 1B 150 0.50 0 0 30 4,500

PL 1A PL 2A 225 0.75 0 0 0 0

PL 1A PL 2B 250 0.83 0 0 0 0

PL 1A Proficient 300 1.00 0 0 0 0

PL 1B PL 1A 0 0.00 0 0 0 0

PL 1B PL 1B 0 0.00 0 0 0 0

PL 1B PL 2A 175 0.58 0 0 15 2,625

PL 1B PL 2B 225 0.75 0 0 15 3,375

PL 1B Proficient 300 1.00 0 0 0 0

PL 2A PL 1A 0 0.00 0 0 0 0

PL 2A PL 1B 0 0.00 0 0 0 0

PL 2A PL 2A 0 0.00 0 0 0 0

PL 2A PL 2B 200 0.67 0 0 5 1,000

PL 2A Proficient 300 1.00 0 0 0 0

PL 2B PL 1A 0 0.00 0 0 0 0

PL 2B PL 1B 0 0.00 0 0 0 0

PL 2B PL 2A 0 0.00 0 0 0 0

PL 2B PL 2B 0 0.00 32 0 0 0

PL 2B Proficient 300 1.00 0 0 0 0

Proficient PL 1A 0 0.00 0 0 0 0

Proficient PL 1B 0 0.00 0 0 0 0

Proficient PL 2A 0 0.00 0 0 0 0

Proficient PL 2B 0 0.00 0 0 0 0

Proficient Proficient 300 1.00 68 20,400 30 9,000

Total 100 20,400 100 20,500

Percent Proficient Year 2 68% 30%

Average Points 204 205

Meet AMO by Status (68% proficient)? Yes No

Meet AMO by Growth (204 Average Points)? Yes Yes
Evaluation of the Growth Model Pilot Project 122

Florida

Florida’s growth model applies to students in grades 3 through 10 using the Developmental Scale 

Scores (DSS) from the Florida Comprehensive Assessment Tests (FCAT) for mathematics and 

reading. The state uses a trajectory model that bases growth targets on the score required for 

proficiency three years after the first year tested (usually, grade 3). To be counted as proficient, 

a student who was not proficient at the baseline must close the gap between his baseline score 

and the proficiency cutoff three grades later by one-third each year. This means that a student 

can only make proficiency by growth for two years because the gap must be fully closed in the 

third year, (i.e., a student must meet or exceed the minimum proficiency score in year 3). A 

student first enrolled in grade 9 must close the gap by half to be counted as “on-track.” The 

cutoffs for grades 3 through 7 are presented in Exhibit B.8.

Exhibit B.8

Florida’s Cutoff Developmental Scale Scores to Be Considered Proficient 

on the FCAT for Grades 3 Through 7

Grade DSS Cutoffs for Reading DSS Cutoffs for Math

3 1,198 1,269

4 1,456 1,444

5 1,510 1,632

6 1,622 1,692

7 1,715 1,786

The illustration in exhibit B.9 below showcases this process for reading. The bold line 

connecting points a, c, f, i, and k represent the FCAT DSS score cutoffs for reading for grades 3 

through 7. The shaded polygons represent the growth increments necessary to be considered ontrack toward proficiency. To illustrate the properties of Florida’s growth model, let us revisit 

student Z and student X. 

Student Z is below proficient. He scored 1,326 in fourth grade (point d), which is below the 

grade-specific cutoff of 1,456. Florida’s growth model would then set growth targets for the 

following two years, fifth grade and sixth grade, by dividing the difference between his fourthgrade score and the seventh-grade cutoff into thirds. In other words, he would be expected to 

close the gap between his fourth-grade score and the seventh-grade cut-off by a third during the 

first year after he was not proficient (fifth grade). In turn, he would be expected to close the gap 

by two-thirds after two years of not being proficient (sixth grade). He would then be expected to 

be proficient after three years (seventh grade). Geometrically, this is drawing a straight line from 

point d at grade 4 to point k at grade 7. This line then sets growth targets at point g and point j 

for grades 5 and 6, respectively. Thus, the growth target for student Z in fifth grade is 1,455 

(point g) and 1,586 in sixth grade (point j). 
Evaluation of the Growth Model Pilot Project 123

Exhibit B.9

Illustration of Florida’s Method for Determining Whether a Student Is 

On-Track Toward Proficiency in Reading

Label a b c d e f g h i j k

Grade 3 4 4 4 5 5 5 6 6 6 7

Score 1,198 1,548 1,456 1,326 1,603 1,510 1,455 1,659 1,622 1,586 1,715

Unlike Arkansas’ trajectory model, Florida’s method is not intentionally designed as a method 

for tracking the progress of proficient students. Florida does not scale growth targets to match 

the pattern of cutoff scores. As a result, this method is very sensitive to the pattern of cutoffs set 

for each grade. This means that if this method were used to draw growth targets for proficient 

students, then the magnitude of the higher expectations would fluctuate between grades, 

depending on grade-by-grade standards. Furthermore, the problem of how far above the cutoff a 

growth target would be for any one grade would greatly depend on which grade was used for the 

base year. 

For example, imagine that student X scored 1,548 (point b) in the fourth grade, which is 92 

points above the cutoff of 1,456 (point c). Let us further assume that for some reason, that score 

was used at a baseline for future growth targets. As a result, that student’s growth targets would 

be represented by the line from point b to point k. By following that line we see that student X 

would have to score 1,603 points to be considered on-track (point e), which is 93 more points 
Evaluation of the Growth Model Pilot Project 124

than the cutoff of 1,510 in the fifth grade (point f). He would not be expected to score as many 

more points in sixth grade, however, because the cutoff scores increase by a large amount 

between the fifth and sixth grades. In the sixth grade, student X would have to score 1,659 

(point h), or only 37 more points than the cutoff of 1,622 (point i). This illustrates that students 

who score above proficient would be subject to inconsistent expectations because the pattern of 

cutoff scores in Florida is not linear, and the model is not designed to mimic the pattern of 

standards from grade to grade or reset expectations.

North Carolina

North Carolina uses a trajectory model to identify growth targets and both End-of-Grade (EOG) 

tests and End-of-Course (EOC) tests for assessing student progress. The state uses a 

Standardized Scale Approach (SSA) to growth which uses the normative distribution of student 

performance in the standard setting year of any test edition as a common basis to build a scale. 

State documents note that this approach is useful for measuring the growth in student 

performance from one year to the next and also adapts well to the changes in curriculum and 

subsequent changes in test editions.

The SSA system uses a time-locked modified z-scale termed a “change scale” or “c-scale.” 

Thus, the c-scale cut score for proficiency on any given test edition at an individual grade level 

remains constant for the life of the scale and test edition regardless of the changes in the 

distribution of test scores that might occur as schools change their instructional methods. The 

state means and standard deviations from the standard setting year are used indefinitely for any 

given test. 

The 2005–06 school year was the standard setting year for the Mathematics EOG tests at grades 

3–8 and the 2002–03 school year was the standard setting year for the Reading EOG tests in 

grades 3–8. North Carolina performs an equating study to set the achievement level cut scores at 

the same time the c-scale is built. 

The trajectory is built based on the student’s performance either the previous year, or on the 

third-grade pretest, whichever is appropriate to the grade in which the student first enters the 

state. Therefore, the following table illustrates the basis for prediction, the targeted test for 

proficiency, the years of trajectory, and the percent of difference between baseline performance 

and proficiency expected by the trajectory based on the year the student is first enrolled in the 

state in a tested grade.
Evaluation of the Growth Model Pilot Project 125

Exhibit B.10 

Grades and Tests Used for Trajectory Growth in North Carolina and the Percent of 

Difference Expected to Be Closed Per Year

Grade of First 

Enrollment

Test Used as the 

Basis for 

Prediction

Test Used as Target for 

Proficiency

Years to 

Proficiency

Percent of 

Difference 

Closed Per 

Step

Steps to 

Proficiency

3 3rd-grade pretest 6th-grade EOG 4 25% 4

4 4th-grade EOG 7th-grade EOG 4 33% 3

5 5th-grade EOG 8th-grade EOG 4 33% 3

6 6th-grade EOG Algebra I or English I EOC 4 33% 3

7 7th-grade EOG Algebra I or English I EOC 4 50% 2

8 8th-grade EOG Algebra I or English I EOC 3 100% 1

The trajectories are built individually by student and separately for reading or mathematics. 

Therefore, a student will have a trajectory based on their baseline mathematics score and the 

proficiency cut score for mathematics separate from reading. In the upper grades, Algebra I is 

the AYP assessment for tenth-grade students and is the trajectory target for math while English I 

is the trajectory target for reading or language arts.

The following table displays the performance expected of students to be counted as on trajectory 

for inclusion in the proposed method of comparing school performance to AMO targets.

For a student who enters in third grade and has a grade 3 pretest:

Year in State-tested Grade

Decrease From Baseline Assessment in Performance 

Discrepancy

1 25% of Original Gap

2 50% of Original Gap

3 75% of Original Gap

4 or more Student Must Be Proficient

For a student who enters in fourth, fifth, or sixth grade:

Year in State-tested Grade

Decrease From Baseline Assessment in Performance 

Discrepancy

1 Baseline, Not On Trajectory

2 33% of Original Gap

3 66% of Original Gap

4 or more Student Must Be Proficient
Evaluation of the Growth Model Pilot Project 126

Therefore, if a subgroup has met its 95 percent participation target but has not met its proficiency 

target, and the subgroup has met its other academic indicator, the process of incorporating the 

growth measure would be:

1) First identify if the student has been in membership the full academic year and is both 

tested and not proficient.

2) These three conditions being met, the number of years the student has been in the state 

will be determined using the historic files from the state’s accountability system.

3) If the student has been in the state (in a tested grade) for four years or more, the student 

will remain non-proficient for comparison to the annual measurable objectives (AMO). 

If the student has been in the state public schools three years or less, the correct baseline 

score will be located (using the table above).

4) The student’s performance on the baseline assessment in the subject of interest will be 

converted to the c-scale.

5) Based on the student’s baseline score and proficiency in the target year, a difference will 

be calculated.

6) The decrease in the difference will be compared against Table 4 above based on the 

number of years in the tested grades in North Carolina.

7) If the student’s performance on the current assessment is equal to or better than the 

minimum from the previous step, include the student in the percent proficient calculation 

to compare against the state’s AMOs.

To illustrate, assume a student enters North Carolina in the fourth grade. The student scores 

below proficient in the current school year in reading. This child’s known test scores are listed 

below.

Grade 3 EOG 4 EOG 5 EOG

Developmental Score Not in NC 229 241

C-scale score -2.68 -1.98

Because the student’s first full year in the state is the fourth-grade year, the student will need to 

be on trajectory to be proficient by the end of the seventh grade and thus on the seventh-grade 

EOG for reading. The developmental score for seventh-grade reading equivalent to proficient is 

252. The associated c-scale score is -1.00.

Because the student was not in the state for the third-grade test, the fourth-grade EOG score will 

be used as the baseline. The difference between the baseline and proficient on the seventh-grade

test in terms of c-scale scores is 1.68 (difference between 2.68 and 1.00). For the current year 

(fifth grade, the second year in the state), the student must perform well enough on the test to 

have 33 percent less difference between the c-scale score for proficiency and his baseline 

(fourth-grade EOG) c-scale score (divide 1.68 by 3 = 0.56). 
Evaluation of the Growth Model Pilot Project 127

For this to be true, the child would need to score at least -2.12 (difference between 2.68 and 

0.56). The child’s actual c-scale score is -1.98 which means the child met the standard to be 

deemed on trajectory for the current year and thus will be included in the percent of students on 

trajectory or proficient for comparison to the AMO for the school as a whole and any subgroups 

to which the child may belong.

Tennessee

Adapted from “Evaluation of the 2005–06 Growth Model Pilot Program,” U.S. Department of 

Education, 2009.

Tennessee uses a projection model to assess student growth. The state uses a student’s history of 

test scores in an equation to project or predict that student’s future score. To complete this 

process, previous cohorts of student scores are used in generating a prediction equation that can 

be applied to the current cohort of students. For example, last year a cohort of sixth-graders in 

Tennessee were tested on the state reading exam. These same sixth-graders also had scores from 

the state reading exam for grades 3, 4, and 5. The scores on the sixth-grade reading test are 

placed in a matrix called Y. The reading scores for grades 3–5 are placed in a matrix called X. 

All the reading scores from grades 3–6 are combined into a design matrix called XY. The 

matrices are used in a statistical procedure to generate a covariance matrix called C with 

submatrices CXX and CXY (CYX = CXY

T

) and CYY. These submatrices are used for various 

statistical functions but primarily in calculations for b = CXX-1CYX to generate the regression 

coefficients b1, b2, ... bN

Projected Score = M

. For example, the projected score is computed using variations of the 

following equation:

Y + b1(X1 – M1) + b2(X2 – M2) + ... + = MY + xi

T

b

where MY, M1, etc. are estimated mean scores for the “future test score” or response variable (Y). 

The previous test scores can also be referred to as the “predictor variables.” To complete 

projected scores in the equation you make the following substitutions:

MY

b

= estimated mean score on test

1, b2, ... bN

X

= regression coefficients used to predict performance

1, X2, ... XN

M

= previous reading scores

1, M2, ... MN

The Tennessee model includes a statewide “average schooling effect,” which is obtained by 

calculating the mean scores for each grade of a particular school and then averaging those means 

over all schools within the state. It is intended to account for the fact that a current school has no 

control over the effectiveness of the schools that their students will attend in the future (thus 

potentially affecting the student’s growth). The average schooling effect assumes that each 

student will have the “average schooling experience” of all Tennessee schools. 

= average school reading scores

Tennessee’s model projects scores for all students, estimating each student’s performance in 

reading and math in three years. Each student’s projection is based upon his or her available test 

scores. For example, student A has reading scores for 2003, 2004, and 2005 whereas student B 
Evaluation of the Growth Model Pilot Project 128

has reading scores for 2003 and 2005. In both cases, projected scores are computed using the 

equation described above using student-specific values.

Exhibit B.11

Illustration of Tennessee’s Method for Determining Whether a Student Is On-Track 

Toward Proficiency in Reading

In Exhibit B.11, student A is currently below proficient (year 1) but projected to be proficient in 

year 4. In Tennessee’s growth model, student A would be considered proficient in year 1 for 

AYP determinations and in each of the succeeding years if he or she continues to be on this 

trajectory to proficient. This is symbolized in how student A’s projection is heading towards the 

performance goal line (the line marked PG in Exhibit B.11). 

Student B is missing data for 2004. Tennessee addresses the issue of missing test scores by 

using the regression coefficients from (b = CXX-1

CYX) and constants in the equation described 

above to fill in the missing test scores. Thus, all relevant student data are included in projecting 

score for a student. Using the available data for student B, a growth trajectory is developed that 

shows a projected decline in performance, though the student is projected to remain above 

proficient. Because student B is projected to remain above proficient by year 4, student B is 

proficient for the current year AYP determinations. If student B’s projection indicated he or she 

would fall below proficient, student B would be considered non-proficient in the current year 

AYP determinations. Performance
Evaluation of the Growth Model Pilot Project 129

Appendix C:

Supplemental Exhibits

Exhibit C.1

Annual Measurable Objectives (AMOs) for Reading and Math, by State, 2006−07 and 

2007−08 School Years

State

Reading Math

2006–07 2007–08 2006–07 2007–08

Alaska 71.48% 77.18% 57.61% 66.09%

Arizona 44.84% 55.86% 38.44% 50.74%

Arkansas 45.49% 53.28% 41.17% 49.58%

Delaware 62.00% 68.00% 41.00% 50.00%

Florida 51.00% 58.00% 56.00% 62.00%

Iowa 67.94% 74.33% 68.10% 74.47%

North Carolina 56.05% 40.85% 68.30% 72.80%

Ohio 71.11% 76.87% 55.31% 64.26%

Tennessee 86.50% 91.00% 77.00% 84.50%

Exhibit reads: In 2006–07, schools in Alaska needed to have 71.48 percent or more of 

the students in each reporting group (all students plus, if present in sufficient numbers,

racial or ethnic groups, LEP students, students with disabilities, and low income students)

scoring at or above the proficiency cut score in order to make AYP. This AMO increased 

to 77.18 percent in 2007–08.

Note: A few states employed a single AMO across all grades. For the states using 

different AMOs for each grade, the AMO’s have been averaged across grades.

Source: Alaska, Arizona, Arkansas, Delaware, Florida, Iowa, North Carolina, Ohio, and Tennessee 

Consolidated State Application Accountability Workbooks.
Evaluation of the Growth Model Pilot Project 130

Exhibit C.2

Number of Eligible Schools and Percentage in Which Math AMO Was Met Because of 

Growth Model Results, by All Students and Racial/Ethnic Reporting Groups, 2007–08

Pilot States

Number of Eligible Schools*

All 

Students White

Black, 

NonHispanic Hispanic

Asian/

Pacific 

Islander

American 

Indian/Alaskan 

Native

All Nine States 1,818 371 1,924 738 47 181

Alaska 74 3 3 0 8 68

Arizona 4 37 59 45 37 99

Arkansas 49 9 77 4 0 0

Delaware** 5 NA NA NA NA NA

Florida 849 106 917 486 1 0

Iowa 71 27 41 28 1 3

North Carolina 198 2 315 104 0 11

Ohio 529 171 487 69 0 0

Tennessee 39 16 25 2 0 0

Percent of Eligible Schools

All Nine States 22% 46% 16% 18% 0% 0%

Alaska 0% 0% 0% NA 0% 0%

Arizona 0% 0% 0% 0% 0% 0%

Arkansas 0% 0% 0% 0% NA NA

Delaware 0% NA NA NA NA NA

Florida 9% 10% 7% 15% 0% NA

Iowa 27% 30% 12% 11% 0% 0%

North Carolina 1% 0% 0% <1% NA 0%

Ohio 53% 81% 49% 77% NA NA

Tennessee 51% 88% 52% 0% NA NA

Exhibit reads: Across all nine states, among the schools in which the “all students” reporting group did 

not reach the mathematics AMO by either status or safe-harbor, that group did reach the AMO in 1,818 

schools when the growth model criteria were applied. The number reaching the AMO by growth 

represented 22 percent of the eligible schools.

* “Eligible schools” means schools that did not make AYP by status or safe-harbor and had the grades 

required to be eligible for growth.

** EDFacts did not include subgroup information for Delaware in 2007–08.

NA: Not applicable due to no eligible schools.

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 131

Exhibit C.3

Number of Eligible Schools and Percentage in Which Math AMO Was Met Because of 

Growth Model Results, by Low-SES, SWD, and LEP Reporting Groups, 2007–08

Pilot States

Number of Eligible Schools*

Economically 

Disadvantaged 

Students

Students With 

Disabilities 

(SWD)

Limited English 

Proficient (LEP) 

Students

All Nine States 2,563 3,018 830

Alaska 81 42 53

Arizona 71 282 157

Arkansas 79 82 8

Delaware** NA NA NA

Florida 1,073 1,184 460

Iowa 143 141 26

North Carolina 363 397 82

Ohio 719 877 43

Tennessee 34 13 1

Percent of Eligible Schools

All Nine States 24% 16% 9%

Alaska 0% 0% 0%

Arizona 0% 0% 0%

Arkansas 0% 0% 0%

Delaware NA NA NA

Florida 9% 2% 11%

Iowa 16% 14% 19%

North Carolina <1% 0% 1%

Ohio 67% 51% 40%

Tennessee 50% 15% 0%

Exhibit reads: Across all nine states, among the schools in which the “economically 

disadvantaged” reporting group did not reach the math AMO by either status or safeharbor, that group did reach the AMO in 2,563 schools when the growth criteria were 

applied. The number reaching the AMO by growth represented 24 percent of the 

eligible schools.

* “Eligible schools” means schools that did not make AYP by status or safe-harbor and 

had the grades required to be eligible for growth.

** EDFacts did not include subgroup information for Delaware in 2007–08.

NA: Not applicable due to no eligible schools.

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 132

Exhibit C.4

Numbers of Schools at Each AYP Status by NCLB School Improvement Status,

by State, 2007−08

Pilot States AYP Status

Identified for 

Improvement or 

Under Corrective 

Action

Planning to 

Restructure or 

Restructuring

Not Identified 

for 

Improvement Total

All Eight*

Making AYP by Status

States

129 13 4,450 4,592

Making AYP by Safe-Harbor 230 43 1,016 1,289

Making AYP by Growth 247 27 962 1,236

Not Making AYP 1,442 836 4,055 6,333

Total 2,048 919 10,483 13,450

Arizona

Making AYP by Status 33 2 985 1,020

Making AYP by Safe-Harbor 12 2 83 97

Making AYP by Growth 3 1 4 8

Not Making AYP 130 35 206 371

Total 178 40 1,278 1,496

Arkansas

Making AYP by Status 22 6 103 131

Making AYP by Safe-Harbor 54 4 311 369

Making AYP by Growth 4 0 48 52

Not Making AYP 133 43 162 338

Total 213 53 624 890

Delaware 

Making AYP by Status 3 0 120 123

Making AYP by Safe-Harbor 0 0 0 0

Making AYP by Growth 0 0 5 5

Not Making AYP 20 15 20 55

Total 23 15 145 183

Florida

Making AYP by Status 13 4 499 516

Making AYP by Safe-Harbor 18 23 75 116

Making AYP by Growth 8 13 132 153

Not Making AYP 305 595 1,595 2,495

Total 344 635 2,301 3,280

Iowa

Making AYP by Status 0 0 678 678

Making AYP by Safe-Harbor 3 0 40 43

Making AYP by Growth 5 1 17 23

Not Making AYP 30 8 316 354

Total 38 9 1,051 1,098

North Carolina

Making AYP by Status 10 0 378 388

Making AYP by Safe-Harbor 55 5 275 335

Making AYP by Growth 0 0 0 0

Not Making AYP 309 44 1,214 1,567

Total 374 49 1,867 2,290

Ohio

Making AYP by Status 20 0 763 783

Making AYP by Safe-Harbor 50 4 109 163

Making AYP by Growth 225 12 736 973

Not Making AYP 452 89 417 958

Total 747 105 2,025 2,877

Tennessee

Making AYP by Status 28 1 924 953

Making AYP by Safe-Harbor 38 5 123 166

Making AYP by Growth 2 0 20 22

Not Making AYP 63 7 125 195

Total 131 13 1,192 1,336

Note: Exhibit C.4 presents the counts underlying Exhibit 23 (“Numbers of Schools Making AYP by Status or Safe-Harbor, 

and Percentage Increase in Schools Making AYP Due to Growth, by NCLB School Improvement Status, 2007−08”). *

Source: U.S. Department of Education, EDFacts

Alaska did not report school improvement status to EDFacts for the 2007–08 school year.
Evaluation of the Growth Model Pilot Project 133

Exhibit C.5

Numbers of Schools at Each AYP Status by School Poverty Concentration, 

by State, 2007−08

Pilot States AYP Status Low Poverty Medium Poverty High Poverty Total

All Nine States

Making AYP by Status 1,733 2,536 547 4,816

Making AYP by Safe-Harbor 213 866 279 1,358

Making AYP by Growth 352 726 168 1,246

Not Making AYP 897 3,883 1,774 6,554

Total 3,195 8,011 2,768 13,974

Alaska

Making AYP by Status 210 0 12 222

Making AYP by Safe-Harbor 66 0 4 70

Making AYP by Growth 0 0 0 0

Not Making AYP 190 0 13 203

Total 466 0 29 495

Arizona

Making AYP by Status 312 482 225 1,019

Making AYP by Safe-Harbor 3 38 56 97

Making AYP by Growth 1 3 4 8

Not Making AYP 31 106 234 371

Total 347 629 519 1,495

Arkansas

Making AYP by Status 4 93 34 131

Making AYP by Safe-Harbor 21 296 52 369

Making AYP by Growth 2 43 7 52

Not Making AYP 3 221 114 338

Total 30 653 207 890

Delaware 

Making AYP by Status 29 90 4 123

Making AYP by Safe-Harbor 0 0 0 0

Making AYP by Growth 0 5 0 5

Not Making AYP 4 47 4 55

Total 33 142 8 183

Florida

Making AYP by Status 255 244 17 516

Making AYP by Safe-Harbor 17 71 28 116

Making AYP by Growth 28 93 32 153

Not Making AYP 316 1,608 571 2,495

Total 616 2,016 648 3,280

Iowa

Making AYP by Status 253 423 2 678

Making AYP by Safe-Harbor 12 28 3 43

Making AYP by Growth 7 15 1 23

Not Making AYP 55 266 33 354

Total 327 732 39 1,098

North Carolina

Making AYP by Status 146 221 17 384

Making AYP by Safe-Harbor 41 250 44 335

Making AYP by Growth 0 0 0 0

Not Making AYP 217 1,031 311 1,559

Total 404 1,502 372 2,278

Ohio

Making AYP by Status 409 354 35 798

Making AYP by Safe-Harbor 49 85 29 163

Making AYP by Growth 313 562 108 983

Not Making AYP 80 527 377 984

Total 851 1,528 549 2,928

Tennessee

Making AYP by Status 115 629 201 945

Making AYP by Safe-Harbor 4 98 63 165

Making AYP by Growth 1 5 16 22

Not Making AYP 1 77 117 195

Total 121 809 397 1,327

Note: Exhibit C.5 presents the counts underlying Exhibit 24 (“Numbers of Schools Making AYP by Status or Safe-Harbor, 

and Percentage Increase in Schools Making AYP Due to Growth, by School Poverty Concentration, 2007−08”).

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 134

Exhibit C.6

Numbers of Schools at Each AYP Status by School Minority Concentration, 

by State, 2007−08

Pilot States AYP Status Low Minority Medium Minority High Minority Total

All Nine States

Making AYP by Status 3,154 1,145 433 4,732

Making AYP by Safe-Harbor 740 408 212 1,360

Making AYP by Growth 895 222 113 1,230

Not Making AYP 1,830 2,854 1,773 6,457

Total 6,619 4,629 2,531 13,779

Alaska

Making AYP by Status 95 65 57 217

Making AYP by Safe-Harbor 28 13 29 70

Making AYP by Growth 0 0 0 0

Not Making AYP 38 74 88 200

Total 161 152 174 487

Arizona

Making AYP by Status 364 452 168 984

Making AYP by Safe-Harbor 4 47 45 96

Making AYP by Growth 1 3 4 8

Not Making AYP 29 106 229 364

Total 398 608 446 1,452

Arkansas

Making AYP by Status 75 37 10 122

Making AYP by Safe-Harbor 268 89 12 369

Making AYP by Growth 28 19 3 50

Not Making AYP 110 152 69 331

Total 481 297 94 872

Delaware 

Making AYP by Status 31 86 6 123

Making AYP by Safe-Harbor 0 0 0 0

Making AYP by Growth 0 5 0 5

Not Making AYP 5 41 9 55

Total 36 132 15 183

Florida

Making AYP by Status 246 182 77 505

Making AYP by Safe-Harbor 29 54 33 116

Making AYP by Growth 31 78 41 150

Not Making AYP 482 1,258 682 2,422

Total 788 1,572 833 3,193

Iowa

Making AYP by Status 646 16 0 662

Making AYP by Safe-Harbor 38 5 0 43

Making AYP by Growth 20 3 0 23

Not Making AYP 250 99 3 352

Total 954 123 3 1,080

North Carolina

Making AYP by Status 282 87 14 383

Making AYP by Safe-Harbor 167 142 29 338

Making AYP by Growth 0 0 0 0

Not Making AYP 403 821 352 1,576

Total 852 1,050 395 2,297

Ohio

Making AYP by Status 726 41 13 780

Making AYP by Safe-Harbor 128 18 16 162

Making AYP by Growth 809 106 57 972

Not Making AYP 449 238 266 953

Total 2,112 403 352 2,867

Tennessee

Making AYP by Status 689 179 88 956

Making AYP by Safe-Harbor 78 40 48 166

Making AYP by Growth 6 8 8 22

Not Making AYP 64 65 75 204

Total 837 292 219 1,348

Note: Exhibit C.6 presents the counts underlying Exhibit 25 (“Numbers of Schools Making AYP by Status or SafeHarbor, and Percentage Increase in Schools Making AYP Due to Growth, by School Minority Concentration, 

2007−08”).

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 135

Exhibit C.7

Numbers of Schools at Each AYP Status by School Urbanicity, by State, 2007−08

Pilot States

Urban 

Schools

Suburban 

Schools

Rural 

Schools Total

All Nine States

Making AYP by Status 984 1,123 2,659 4,766

Making AYP by Safe-Harbor 343 218 801 1,362

Making AYP by Growth 209 492 531 1,232

Not Making AYP 2,140 1,733 2,643 6,516

Total 3,676 3,566 6,634 13,876

Alaska

Making AYP by Status 21 4 193 218

Making AYP by Safe-Harbor 13 2 55 70

Making AYP by Growth 0 0 0 0

Not Making AYP 55 3 142 200

Total 89 9 390 488

Arizona

Making AYP by Status 427 194 365 986

Making AYP by Safe-Harbor 44 19 34 97

Making AYP by Growth 5 1 2 8

Not Making AYP 189 56 120 365

Total 665 270 521 1,456

Arkansas

Making AYP by Status 13 5 104 122

Making AYP by Safe-Harbor 60 30 279 369

Making AYP by Growth 14 5 31 50

Not Making AYP 86 25 220 331

Total 173 65 634 872

Delaware

Making AYP by Status 13 57 53 123

Making AYP by Safe-Harbor 0 0 0 0

Making AYP by Growth 3 1 1 5

Not Making AYP 16 24 15 55

Total 32 82 69 183

Florida

Making AYP by Status 107 284 118 509

Making AYP by Safe-Harbor 45 43 28 116

Making AYP by Growth 35 92 24 151

Not Making AYP 681 1,150 609 2,440

Total 868 1,569 779 3,216

Iowa

Making AYP by Status 66 41 565 672

Making AYP by Safe-Harbor 9 2 32 43

Making AYP by Growth 1 2 20 23

Not Making AYP 123 25 206 354

Total 199 70 823 1,092

North Carolina

Making AYP by Status 55 60 281 396

Making AYP by Safe-Harbor 64 52 222 338

Making AYP by Growth 0 0 0 0

Not Making AYP 449 206 952 1,607

Total 568 318 1,455 2,341

Ohio

Making AYP by Status 71 332 381 784

Making AYP by Safe-Harbor 34 46 83 163

Making AYP by Growth 142 389 442 973

Not Making AYP 421 233 305 959

Total 668 1,000 1,211 2,879

Tennessee

Making AYP by Status 211 146 599 956

Making AYP by Safe-Harbor 74 24 68 166

Making AYP by Growth 9 2 11 22

Not Making AYP 120 11 74 205

Total 414 183 752 1,349

Note: Exhibit C.7 presents the counts underlying Exhibit 26 (“Numbers of Schools Making AYP by Status or SafeHarbor, and Percentage Increase in Schools Making AYP Due to Growth, by School Urbanicity, 2007−08”).

Source: U.S. Department of Education, EDFacts.
Evaluation of the Growth Model Pilot Project 136

Exhibit C.8

Percentage of Schools Meeting the AMO Using Growth-Only and Making AYP 

by Other Means After Growth-Only Is Applied, by State, 2007–08

Pilot States

Making AYP 

by Status

Making AYP 

by Safe 

Harbor

Making AYP 

by Growth

Not Making 

AYP Total

All Nine States

Not Making AYP by Growth-Only 1,415 771 202 5,220 7,608

Making AYP by Growth-Only 3,080 548 1,042 930 5,600

Total 4,495 1,319 1,244 6,150 13,208

Alaska

Not Making AYP by Growth-Only 12 19 0 83 114

Making AYP by Growth-Only 206 51 0 119 376

Total 218 70 0 202 490

Arizona

Not Making AYP by Growth-Only 517 75 7 330 929

Making AYP by Growth-Only 440 17 1 19 477

Total 957 92 8 349 1,406

Arkansas

Not Making AYP by Growth-Only 59 108 27 274 468

Making AYP by Growth-Only 65 248 25 54 392

Total 124 356 52 328 860

Delaware 

Not Making AYP by Growth-Only 47 0 4 50 101

Making AYP by Growth-Only 75 0 1 4 80

Total 122 0 5 54 181

Florida

Not Making AYP by Growth-Only 211 91 111 2,419 2,832

Making AYP by Growth-Only 305 25 42 75 447

Total 516 116 153 2,494 3,279

Iowa

Not Making AYP by Growth-Only 279 30 15 299 623

Making AYP by Growth-Only 363 13 8 52 436

Total 642 43 23 351 1,059

North Carolina

Not Making AYP by Growth-Only 110 279 0 1,215 1,604

Making AYP by Growth-Only 145 42 0 32 219

Total 255 321 0 1,247 1,823

Ohio

Not Making AYP by Growth-Only 10 12 23 364 409

Making AYP by Growth-Only 690 143 958 563 2,354

Total 700 155 981 927 2,763

Tennessee

Not Making AYP by Growth-Only 170 157 15 186 528

Making AYP by Growth-Only 791 9 7 12 819

Total 961 166 22 198 1,347

Note: Exhibit C.8 presents the counts underlying Exhibits 31 and 32 (“Percentage of Schools Meeting AMO Using Only the 

Growth Model On-Track Indicator, by Standard EDFacts AYP Classification and State, 2007–08” and “Percentage of 

Schools Meeting the AMO Using Growth-Only and Making AYP by Other Means After Growth-Only is Applied, by State, 

2007–08,” respectively).

Source: U.S. Department of Education, EDFacts and the Alaska, Arizona, Arkansas, Delaware, Florida, Iowa, North Carolina, Ohio, and 

Tennessee state departments of education.
Evaluation of the Growth Model Pilot Project 137 

Appendix D: 

Derivation of the Generic Projection Model Rule for Identifying 

On-Track Students 

The slope of the projection model line in Exhibit 45 and subsequent exhibits is initially 

surprising in its stark contrast with alternative models. This appendix explains the derivation of 

the line in these figures. 

The prediction equations used by the projection model are shown below. Note that they are 

simplified due to the standardization of the North Carolina data in the construction of a “generic” 

dataset. 

ܴ෠

௚ାே ൌ ߚመ

ோଵ

መߚ ൅ ଵି௚ܴ כ

ோଶ

௚ܴ כ

ܯ௚෡ାே ൌ ߚመ

ெଵ

መߚ ൅ ଵି௚ܯ כ

ெଶ

௚ܯ כ

The projection model line is easily calculated from these prediction equations. For example, the 

predicted reading score in some future grade ݃൅ܰ is ܴ෠

௚ାே. This is compared with the 

proficient cut score at that grade, ܿ௚ାே. Thus, the projection model decision rule for Reading is 

defined by the following: 

If ߚመ

ோଵ

መߚ ൅ ଵି௚ܴ כ

ோଶ

כ ܴ ൒ ௚ܿ௚ାே, then the student is on track. 

The framework displays this decision rule on a plot of ܴ௚ on ܴ௚ିଵ. Thus, solving for the vertical 

axis, ܴ௚, the decision line can be written in slope-intercept form as follows: 

ܴ௚ ൌ ቆെ ߚመ

ோଵ

כ

መߚ

ோଶ

כ ቇ ܴ௚ିଵ ൅ ቆܿ௚ାே

መߚ

ோଶ

ቇ כ

The contrast between the projection model line and the positive-sloped trajectory and transition 

matrix model decision lines is revealed by the simple fact that regression coefficients in this 

framework are positive, leading to a projection model line that is negative. Exhibit D.1 

overviews the regression coefficients and corresponding slopes for the North Carolina data. The 

regression line in the framework is plotted from the Math dataset that uses prediction grades 4 

and 5 to project to grade 8. As Exhibit D.1 shows, the use of an alternative grade would not 

change the fundamental contrast between projection and trajectory models, and in most cases the 

contrast would be increased. 
Evaluation of the Growth Model Pilot Project 138 

Exhibit D.1 

Regression Coefficients and Framework Slopes for North Carolina Data 

Subject Prediction Grades (݃െ1 and ݃) Projection 

Grade ߚመ

௝ଵ

መߚ כ

௝ଶ

כ Slope ቀെ ఉ෡ೃభ

כ

ఉ෡ೃమ

ቁ כ

Reading 2 and 3 6 0.286 0.543 -0.527 

Reading 3 and 4 7 0.388 0.491 -0.688 

Reading 4 and 5 8 0.378 0.469 -0.806 

Reading 5 and 6 8 0.383 0.487 -0.787 

Reading 6 and 7 8 0.407 0.481 -0.846 

Math 3 and 4 7 0.366 0.501 -0.730 

Math 4 and 5 8 0.336 0.512 -0.657 

Math 5 and 6 8 0.318 0.562 -0.566 

Math 6 and 7 8 0.364 0.539 -0.675 

The Department of Education’s mission is to promote student achievement and 

preparation for global competitiveness by fostering educational excellence and ensuring equal access. 

www.ed.gov

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