independent case study equals mathematics december 2014 · pdf file2 equals math curriculum...

Independent Case Study Equals® MathematicsDecember 2014

For Equals Mathematics product information www.ablenetinc.com/equals

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Longitudinal Summary of Results Students using Equals Mathematics over three years saw 112% average gains

across students with mild, moderate, or severe/profound disabilities

Updated Case StudyEquals Mathematics is a K-12 standards-based curriculum that employs best practice math instruction as identified in re-search literature. A suburban Midwestern school district conducted an independent 3-year study of the effectiveness of the Equals Mathematics curriculum. Two additional studies were started in subsequent years to measure gains across two new groups of students. The three studies are labeled Group 1, Group 2, and Group 3.

Research MethodologyIn all instances, instruction in Equals Mathematics was provided to participating students daily for 40-minutes in small group settings. Baseline data was collected before Equals Mathematics instruction began in August and progress was then assessed each spring. Baseline and end-of-year data was recorded using the Equals Assessment protocol. The pre- and post-Equals Assessment data was collected by the staff. The Assessment staff was trained in the Equals Assessment admin-istration, but were not affiliated with the students they assessed.

Analysis Summary STUDY 1Students using Equals Mathematics over 3-years were assessed using the pre- and post-Equals Assessment. Initial assess-ment was completed in September 2011 and compared to the post-assessment captured in April 2014. All students using Equals Mathematics over a three-year period of time increased their math skills and showed no regression.

0

50

100

150

200

250Sep 2011

Apr 2014

Level 3Mild

Level 2 Moderate

Level 1Severe/Profound

Group 1 - 3 Year Results

22.5

77.263.1

135.7 133.2

236.7

*For full data set, see full Equals White Paper

Average gains of 112%

STUDY 3All 3 groups of first year Equals Mathematics students demonstrated an increase in math skills and no student regressed using

the program.

STUDY 2Group 2: All students using Equals Mathematics over a two-year period of time increased their math skills and showed no regression.

0

20

40

60

80

100

Level 2 Moderate

Level 1 Severe/Profound

Aug 2012

Apr 2014

30

50

70

90

10


22.8

55.7

34

91

0

50

100

150

200

250Sep 2011

Apr 2013

Level 3 Mild

Level 2Moderate



22.5

65 63.1

117.7133.2

216

0

20

40

60

80

Level 2 Moderate


10

30

50

70Apr 2013

Aug 2012


22.7

40.934

69

0

50

100

150

200

Apr 2012

Sep 2011

Level 3 Mild

Level 2 Moderate



22.5

46.4

133.2

179.3

63.1

86.1

0

30

60

90

120

150

Level 3 Mild

Level 2 Moderate


Apr 2014

Sep 2013


16.325.5

92.1

138.2

51

75.7



About Equals Mathematics Equals Mathematics was developed by drawing from general education math standards from around the country and best practice math instruction, learning strategies, and pedagogy from a wide variety of resources and research materials. These included National Council of Teachers of Mathematics recommendations and periodicals, publications from other sources (books, papers, journals, and periodicals), major math publishers, math instructors, and special educators. In addition, action research was conducted in large and small school districts across the United States, with K-12 students who have mild, moderate, and significant disabilities.

Equals Includes:

The Equals Technology Lesson Center (ETLC) is a web-based tool for accessing Equals lesson materials for use with any interactive white-board, tablet, desktop, or desktop with projector. The ETLC provides visual materials appropriate for group instruction, problem solving, and demonstration for every Equals lesson.

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Equals Manipulative Kit Materials

over 100 manipulatives

12 Teacher Guides with 325+ lessons

More than 700 Vocabulary & Content Cards

25 Math Content & Theme Posters

2 Assessment Flipbooks & 10 Student Response Booklets

27 Student Workmats

Equals Overview

Mathematics Curriculum forStudents of Differing Abilities

Equals Overview

Members Only Teacher Portal

(includes assessment)

Thousands of additional resources including training modules, chapter tests, data forms, and classroom videos of the

Equals curriculum in action.

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1

Executive Summary

The Developmental Learning Program in Midlothian, Illinois has been independently collecting data on the

effectiveness of Equals Mathematics Curriculum for three years. The study focused on students, K-‐8, with mild,

moderate, and severe disabilities including autism, cognitive disabilities, and multiple disabilities.

At the end of Year 1, results indicated that 72 of 72 students increased their math skills as measured by the Equals assessment. No regression was found. Students increased their math skills at an average of 38%. At the end of Year 2, 58 of the 58 students remaining in the study increased their math skills at the same rate with no regression reported. Over two years, the 58 remaining students increased their math skills at an average of 83%. Year 3 was completed with the remaining 43 students demonstrating 112% progress in math skills from 2011-2014.

Thirty new students were included in the study in Year 2. The remaining 26 students continued to make progress in

the second year at a slightly slower rate than the previous year. This group differs from the original study group in that

there are students in the group from Levels 1 and 2, but not Level 3. Finally, 34 new students were added to the study

in the third year. Again, the group make-up differs from the original study group in that there are fewer Level 3

students than Level 1 and Level 2 students. All students in this group made progress but at a lower rate than the other

two groups cited in the study. Conclusions can be drawn from this study, which indicate the same group of students

achieved a strong gain in math skills across three years with progress shown with three different groups of students.

No regression has been reported in any of the groups.

Outline of Materials Page

A. Addendum: Year 3 Research Study Findings

1. Summary 3

2. Data: 3 years 4

3. Data: new students and 2nd year students 9

B. Addendum: Year 2 Research Study Findings

1. Summary 10

2. Data: 2 years 15

3. Data: new students and 2nd year students 19

C. Addendum: Year 1 Research Study Findings

1. Description 23

2. Summary 28

3. Data 38

Addendum

2

Equals Math Curriculum Study, Year 3: 2013-‐2014

Ben Satterfield, Ed.D.

The study of the Equals math curriculum at the Developmental Learning Program (DLP) has recently completed its

third year. In all, 103 students with disabilities in grades K-8 (ages 5-14) and 21 teachers took part in the study

during the 2013-14 school year. Equals math instruction was provided to participating students each day for 40

minute sessions in groups of 3 to 8 students from September through May. Once again, students were tested in

the fall and the spring using the Equals assessment protocol. Spring scores were compared to fall scores to

determine the level of progress for each participating student. For the third consecutive year, every student in the

project demonstrated progress in math with no regression reported.

Of the 72 students who started using Equals in 2011, 43 have continued to participate for three consecutive years.

There are 26 additional students who completed their second year in the program and 34 additional students who

were included for the first time this year. The disabilities of the participating students included cognitive disabilities

(mild, moderate, and severe), autism, emotional disorder, multiple disabilities, and other health impairments.

Among students participating for the third year, every student demonstrated progress in math again this year, as

they had for the previous two years. Taken as a group, students in their third year continued to progress, but at a

slower rate than the previous two years. This was partially attributed to the fact that the concepts encountered

are more challenging as students advance through the Equals curriculum. It was also observed that several

students who had demonstrated rapid progress during years 1 and 2, encountered personal issues that were a

distraction that impacted their school performance as a whole.

All of the 26 second-year students who have been in the program made progress in math again this year. The

mean difference test scores from August to May for this group were 18.5 points. This was about 8 points less than

the scores for the same group last year. Again, this appears to have been a general pattern demonstrated by all

students in this group. Groups whose mean average scores on the Equals protocol exceeded those of the group as

a whole included: (1) students with autism and (2) students with moderate intellectual disabilities (level 2).

In year three, there were 34 students entering the project for the first time. All students in this group made

progress in math using Equals. The first year mean scores were lower than the score of students entering the

program for the first time in the previous two years. In analyzing this comparison, it was determined that the

make-up of the students in the program has changed in two important ways. First, the diversity of the students is

increasing as students with a greater range of disabilities are entering the DLP program. Second, a greater

proportion of those newly entering students are considered to have a severe (level 1) or moderate (level 2)

intellectual disability, while a lesser proportion are students with mild intellectual disability (level 3).

Overall, the most striking result drawn from this year's data is the fact that every student in this study continued to

make progress in developing his or her math skills. Over the three year period, several sub-groups have

consistently shown above average progress. These include: (1) students with mild intellectual disability (level 3

students), (2) male students, (3) students with autism, (4) students with low socio-economic-status (SES), and (5)

students from Hispanic backgrounds.

3

Teachers at DLP attribute the progress to the theoretical construct and teaching strategies of the Equals curriculum. One teacher observed:

“Equals has been an interesting journey. I have taught 3 different chapters so far and it has gone well because of the well thought-out organization and lay-out of the lessons and theory behind the process. Being a teacher with a bit more “experience” than some of the other teachers, I have learned many new ways of presenting information and developed many new skills through Equals Math.”

Others at DLP have remarked upon the Equals manipulatives and supporting materials that engage students in learning math and allow them to build upon skills and strengths. The new online Equals Tech Center provides teachers more ways to make math more visual and concrete:

“Equals Math has been a great addition to the curriculum at DLP. I have been able to teach my students concepts that I never thought I could because of the way the lessons build on top of each other. More importantly the students have been able to retain these concepts and build on their skills. The new addition of the Equals Tech Center has given us another way for the students to practice the concepts they are working on.”

The continuing student progress is even more impressive when one considers how challenging it can be to teach math to students with the combinations of disabilities and cognitive levels of the students who attend the DLP. Teachers report feeling empowered and encouraged because of the way the Equals curriculum provides a framework for presenting math instruction in which every student can become engaged and learn. A third teacher commented:

“I feel that Equals has let students with even the most significant needs access mathematics in a way they have never been able to do before. It's not just about counting, time, and money any longer. The students are using vocabulary and strategies I don't think any of us could ever have imagined. Being able to use the chapter approach has allowed each teacher to micro-focus and become an expert in teaching [to] a multitude of [student] needs [within] the same content area in both full and small groups.”

The results from the third year of this study indicate that progress is being made by students participating in this

prescribed instruction using Equals Math curriculum. It is remarkable that every student in this study has

demonstrated progress each year in the program. Teachers’ reports of gaining a particular sense of empowerment

in regard to using Equals with students who have significant cognitive and physical challenges, add credibility to

the fact that the math instructional methodologies and differentiation in Equals are successful tools for our most

challenged learners. Equals has proven to be a highly effective math curriculum for all students with mild,

moderate and severe disabilities.

4

EQUALS MATH CURRICULUM DLP STUDY –

Group 1 YEAR 3 DATA

N Sept. 2011 Apr. 2012

ΔYR1 p-Value (0.05) Δ Year 1

Apr. 2013 ΔYR2 p-Value (0.05) Δ Year 2

April 2014


ΔYR1-3 p-Value (0.05) Δ

Years 1- 3

All 3 Yr. Participants 43 65.6163 93.6163 28.0000 0.0000 122.3721 28.7558 0.0000 139.1628 16.7907 0.0000 73.5465 0

Level

N Sept. 2011 Apr. 2012



April 2014



Years 1- 3

Level 1 13 22.4615 46.3846 23.9231 0.0000 65.0385 18.6538 0.0001 77.1923 12.1538 0.0000 54.7308 0.0000

Level 2 21 63.0952 86.1429 23.0476 0.0000 117.7143 31.5714 0.0000 135.7143 18.0000 0.0000 72.6190 0.0000

Level 3 9 133.38333 179.2778 45.4444 0.0009 216.0556 36.7778 0.0004 236.7222 20.6667 0.0023 102.8889 0.0000

Category

N Sept. 2011 Apr. 2012



April 2014



Years 1- 3

Autism 20 68.5750 103.175

0 34.6000 0.0000 1333.4000 30.2250 0.0000 151.2500 17.8500 0.0000 82.6750 0.0000

Cognitive Impairment 14 63.1071 88.3214 25.2143 0.0000 117.5357 29.2143 0.0000 131.1429 13.6071 0.0000 68.0357 0.0000

Developmental Delay 1 22.0000 40.0000 18.0000 NA 66.0000 26.0000 NA 88.0000 22.0000 NA 66.0000 NA

Emotional Disorder 1 194.5000

225.5000 31.0000 NA 240.0000 14.5000 NA 285.0000 45.0000 NA 90.5000 NA

Multiple 6 45.6667 61.1667 15.5000 0.0028 87.5833 26.4167 0.0251 103.0000 15.4167 0.0037 57.3333 0.0049

Other Health Impairment 1 76.0000 93.0000 17.0000 NA 117.0000 24.0000 NA 132.0000 15.0000 NA 56.0000 NA

5

Gender

N Sept. 2011 Apr. 2012



April 2014



Years 1- 3

Female 8 73.1875 89.4375 16.2500 0.0000 117.3125 27.8750 0.0005 134.1250 16.8125 0.0012 60.9375 0.0000

Male 35 63.8857 94.5714 30.6857 0.0000 123.5286 28.9571 0.0000 140.3143 16.7857 0.0000 76.4286 0.0000

SES/ Lunch Program

N Sept. 2011 Apr. 2012



April 2014



Years 1- 3

Free 32 66.9688 97.1563 30.1875 0.0000 128.0469 30.8906 0.0000 144.5625 16.5156 0.0000 77.5938 0.0000

Reduced 2 34.5000 56.2500 21.7500 NA 78.5000 22.2500 NA 90.7500 12.2500 NA 56.2500 NA

Paid 9 67.7222 89.3333 21.6111 0.0000 111.9444 22.6111 0.0006 130.7222 18.7778 0.0042 63.0000 0.0000

Ethnicity N Sept. 2011 Apr. 2012



April 2014



Years 1- 3

African- American 15 66.2000 95.1000 28.9000 0.0000 122.7000 27.6000 0.0000 137.7000 15.0000 0.0001 71.5000 0.0000

Asian 1 54.5000 78.5000 24.0000 NA 99.5000 21.0000 NA 105.0000 5.5000 NA 50.5000 NA

Hispanic 9 58.0556 103.111

1 45.0556 0.0011 134.8333 31.7222 0.0019 149.5000 14.6667 0.0006 91.4444 0.0005

White 18 69.5278 88.4722 18.9444 0.0000 117.1389 28.6667 0.0000 137.1111 19.9722 0.0000 67.5833 0.0000

Level 1 N Sept. 2011

Apr. 2012



April 2014



Years 1- 3

Autism 3 17.1667 54.3333 37.1667 0.1064 73.8333 19.5000 0.2816 87.0000 13.1667 0.1811 69.8333 0.0228


Developmental Delay 1 22.0000 40.0000 18.0000 NA 66.0000 26.0000 NA 88.0000 22.0000 NA 66.0000 NA

Emotional Disorder 0 - - - - - - - - - - - -

Multiple 2 14.2500 24.7500 10.5000 NA 36.0000 11.2500 NA 45.7500 9.7500 NA 31.5000 NA

Other Health Impairment 0 - - - - - - - - - - - -

6


Apr. 2012



April 2014



Years 1- 3

Autism 13 62.5000 91.5000 26.6923 0.0009 122.0000 30.5000 0.0000 140.3077 18.3077 0.0000 75.5000 0.0000


Developmental Delay 0 - - - - - - - - - - - -

Emotional Disorder 0 - - - - - - - - - - - -

Multiple 4 61.3750 79.3750 18.0000 0.0694 113.3750 34.0000 0.0275 131.6750 18.2500 0.0187 70.2500 0.0046

Other Health Impairment 1 76.0000 93.0000 17.0000 NA 117.0000 24.0000 NA 132.0000 15.0000 NA 56.0000 NA


Apr. 2012



April 2014



Years 1- 3

Autism 4 119.3750 177.000

0 58.3750 0.0480 215.1250 37.3750 0.1090 235.0000 19.8750 0.0599 188.2917 0.0399

Cognitive Impairment 4 133.125

169.2500 36.1250 0.0102 211.0000 41.7500 0.0136 236.3750 15.3750 0.0115 93.2500 0.0074

Developmental Delay 0 - - - - - - - - - - - -

Emotional Disorder 1 194.5000

222.5000 31.0000 NA 240.0000 14.5000 NA 285.0000 45.0000 NA 90.5000 NA

Multiple 0 - - - - - - - - - - - -

Other Health Impairment 0 - - - - - - - - - - - -

7

LEVEL 1 N Sept. 2011

Apr. 2012



April 2014



Years 1- 3


Asian 0 - - - - - - - - - - - -

Hispanic 4 19.6250 51.3750 31.7500 0.0128 71.5000 20.1250 0.7780 83.6750 12.1250 0.0427 64.0000 0.0111

White 5 25.1000 39.2000 14.1000 0.0073 55.8000 16.6000 0.0065 69.8000 14.0000 0.0055 44.7000 0.0032


Apr. 2012



April 2014



Years 1- 3


Asian 1 54.5000 78.5000 24.0000 NA 99.5000 21.0000 NA 105.0000 5.5000 NA 50.5000 NA

Hispanic 1 37.5000 58.5000 21.0000 NA 88.0000 29.5000 NA 96.5000 8.5000 NA 59.0000 NA

White 10 61.8000 81.0500 19,25 0.0008 116.3500 35.3000 0.0000 136.2500 19.9000 0.0000 74.4500 0.0000


Apr. 2012



April 2014



Years 1- 3

African- American 2 145.0000

181.7500 36.7500 NA 219.5000 37.7500 NA 229.7500 10.2500 NA 84.7500 NA

Asian 0 - - - - - - - - - - - -

Hispanic 4 101.6250 166.000

0 64.3750 0.0058 209.8750 43.8750 0.0068 228.6250 18.7500 0.0067 127.0000 0.0015

White 3 169.3333 195.333

3 26.0000 0.0318 222.0000 26.6667 0.1351 252.1667 30.1667 0.0639 82.8333 0.0166

8


Apr. 2012



April 2014



Years 1- 3

Female 0 - - - - - - - - - - - -

Male 13 22.4615 46.3846 23.9231 0.0004 65.0385 18.6538 0.0002 77.1923 12.1539 0.0009 54.7308 0.0001


Apr. 2012



April 2014



Years 1- 3

Female 7 59.0000 75.2143 16.2143 0.0000 104.6429 29.4286 0.0000 119.8571 15.2143 0.0000 60.8571 0.0000

Male 14 65.1429 91.6071 26.4643 0.0002 124.2500 32.6429 0.0000 143.6429 19.3929 0.0000 78.5000 0.0000


Apr. 2012



April 2014



Years 1- 3

Female 1 172.5000 189.000

0 16.5000 NA 206.0000 17.0000 NA 234.0000 28.0000 NA 61.5000 NA

Male 8 129.0000 178.062

5 49.0625 0.0003 217.3125 39.2500 0.0001 237.0625 19.7500 0.0024 108.0625 0.0000

9

EQUALS MATH CURRICULUM DLP STUDY

Group 2 YEAR 2 DATA

(includes data from 2012 – 2014)

GROUP N August (2012)

April (2013)

Mean Difference Year 1

p-Value (0.05) Δ Year

1 April

(2014)


p-Value (0.05) Δ Year 2

Mean Difference 2

Years p-Value (0.05) Δ Years 1&2

Students in Yr 2 26 28.4038 54.9231 26.5192 0.0000 73.3846 18.4615 0.0000 46.1154 0.0127

Instructional Level N

August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Level 1 13 22.7692 40.8846 18.1154 0.0000 55.7308 14.8462 0.0001 34.3077 0.1489

Level 2 13 34.0385 68.9615 34.9231 0.0000 91.0385 22.0769 0.0003 57.9231 0.0919

Level 3 0 - - - - - - - - -

Category N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Autism 10 31.1000 58.6500 27.5500 0.0007 80.6500 22.0000 0.0013 51.3000 0.3748

Cog. Disability 10 20.0000 45.5500 25.5500 0.0012 62.7500 17.2000 0.0020 42.7500 0.0880

Devel. Delay 6 37.9167 64.3333 26.4167 0.0001 79.0000 14.6667 0.0013 43.0833 0.0149

Gender N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Female 7 23.4286 50.2857 26.8571 0.0052 68.7143 18.4286 0.0026 47.0000 0.0003

Male 19 30.2368 56.6136 26.3947 0.0000 75.1053 18.4737 0.0000 45.7895 0.0000

Ethnicity N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


African-American 10 26.5000 52.0000 25.5000 0.0001 70.6000 18.6000 0.0012 45.9000 0.0002

Hispanic 4 27.2500 54.8750 27.6250 0.1861 69.1250 14.2500 0.0651 41.8750 0.1228

10

White 12 30.3750 57.3750 27.0000 0.0001 77.1250 19.7500 0.0023 47.7083 0.0000

SES N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Free 21 25.3095 51.9286 26.6190 0.0000 68.5000 16.5714 0.0000 44.0238 0.0000

Partial 0 - - - - - - - - -

Paid 5 41.4000 67.5000 26.1000 0.0043 93.9000 26.4000 0.0895 54.9000 0.0250

Level 1 N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Autism 5 24.9000 41.4000 16.5000 0.0048 60.5000 19.1000 0.0520 39.1000 0.0187

Cog. Disability 6 17.4167 35.3333 17.9167 0.0130 47.4167 12.0833 0.0070 30.0000 0.0020

Devel. Delay 2 33.5000 56.2500 22.7500 NA 68.7500 12.5000 NA 32.2500 NA


April (2013)



1 April

(2014)



Mean Difference 2


Autism 5 37.3000 75.9000 38.6000 0.0214 100.800

0 24.9000 0.1170 63.5000 0.0027

Cog. Disability 4 23.8750 60.8750 37.0000 0.0804 85.7500 24.8740 0.1079 61.8750 0.0914

Devel. Delay 4 40.1250 68.3750 28.2500 0.0011 84.1250 15.5750 0.0109 44.0000 0.0019


April (2013)



1 April

(2014)



Mean Difference 2



Hispanic 2 21.0000 34.0000 13.0000 NA 43.0000 9.0000 NA 22.0000 NA

White 6 25.8333 46.7800 20.9167 0.0031 64.3333 17.5833 0.0072 38.4167 0.0001

11


April (2013)



1 April

(2014)



Mean Difference 2



Hispanic 2 33.5000 75.7500 42.2500 NA 95.2500 19.5000 NA 61.7500 NA

White 6 34.9167 68.0000 33.0833 0.0143 89.9167 21.9167 0.0889 55.0000 0.0054

Female N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Level 1 4 18.6250 38.1250 19.5000 0.0178 58.3750 20.2500 0.0595 39.7500 0.0377

Level 2 3 29.8330 66.5000 36.6667 0.2118 82.5000 16.0000 0.1589 52.6667 0.1279

Level 3 0 - - - - - - - - -

Male N August (2012)

April (2013)



1 April

(2014)



Mean Difference 2


Level 1 9 24.6111 42.1111 17.5000 0.0003 54.5556 12.4444 0.0007 31.8889 0.0000

Level 2 10 35.3000 69.7000 34.4000 0.0001 93.6000 23.9000 0.0019 58.3000 0.0000

Level 3 0 - - - - - - - - -

12

EQUALS MATH CURRICULUM DLP STUDY

Group 3 YEAR 1 DATA

(includes data from 2013-2014)

ALL N Aug-13 Apr-14 Δ p-Value (0.05)

All Students 34 42.1176 63.4706 21.3529 0.0000

Level N Aug 13 Apr-14 Δ p-Value (0.05)

Level 1 17 16.3235 25.5000 9.177 0.0000

Level 2 10 50.9500 75.7000 24.750 0.0002

Level 3 7 92.1429 138.2413 46.098 0.0005

Category N Aug 13 Apr-14 Δ p-Value (0.05)

Autism 8 29.8750 50.1250 20.2500 0.0120

Cognitive Impairment 11 36.9545 51.2727 14.3182 0.0152

Developmntal Delay 11 39.9091 69.1818 29.2727 0.0011

OHI 2 17.7500 24.2500 6.5000 NA

Phys. Impaiment 1 175.5000 202.0000 26.5000 NA

Speech Lang Disorder 1 136.5000 181.5000 45.0000 NA

Gender N Aug 13 Apr-14 Δ p-Value (0.05)

Female 9 41.2222 57.7778 16.5555 0.0443

Male 25 42.4400 65.5200 23.0800 0.0000

SES N Aug 13 Apr-14 Δ p-Value (0.05)

Free 24 46.3750 68.1458 21.7708 0.0000

Reduced 2 73.2500 103.7500 30.5000 NA

Paid 8 21.5625 39.3750 17.8125 0.0880

Ethnicity N Aug 13 Apr-14 Δ p-Value (0.05)

African American 11 44.8182 64.5909 19.7727 0.0019

Hispanic 8 35.6875 48.0000 12.3125 0.0120

White 14 45.4286 73.9643 28.5357 0.0002

Asian 1 17.5000 28.0000 10.5000 NA

Level 1 N Aug 13 Apr-14 Δ p-Value (0.05)

Autism 4 17.5000 28.3750 10.8750 0.0384

Cog Imp 8 14.3125 20.7500 6.4375 0.0095

Devel Delay 3 19.1667 35.1667 16.0000 0.2325

OHI 2 17.7500 24.2500 6.5000 NA

13


Autism 4 42.2500 54.1250 11.8750 0.0578

Cog Imp 2 99.7500 126.5000 26.7500 NA

Devel Delay 4 35.2500 54.1250 18.8750 0.0274


Cog Imp 1 92.5000 145.0000 52.5000 NA

Devel Delay 4 60.1250 109.7500 49.6250 0.0321

Phys Impairment 1 175.5000 202.0000 26.5000 NA

Speech Lang Disorder 1 136.5000 181.5000 45.0000 NA


African-American 6 20.0000 28.4167 8.4167 0.0042

Hispanic 5 9.3000 15.5000 6.2000 0.0317

White 5 18.7000 31.5000 12.8000 0.0441

Asian 1 17.5000 28.0000 10.5000 NA


African-American 3 71.0000 95.1667 24.1667 0.1204

Hispanic 2 31.7500 52.2500 20.5000 NA

White 5 46.6000 73.4000 26.8000 0.0321


African-American 2 80.0000 127.2500 47.2500 NA

Hispanic 1 175.5000 202.0000 26.5000 NA

White 4 77.3750 127.7500 50.3750 0.0272

Female N Aug 13 Apr-14 Δ p-Value (0.05)

Level 1 6 16.1667 23.3333 7.1666 0.0006

Level 2 2 68.7500 99.2500 30.5000 NA

Level 3 1 136.5000 181.5000 45.0000 NA

MALES N Aug 13 Apr-14 Δ p-Value (0.05)

Level 1 11 16.4091 26.6818 10.2727 0.0006

Level 2 7 48.5000 68.1429 19.6429 0.0004

Level 3 7 77.2857 123.9286 46.6429 0.0005

14

Addendum

Equals Math Curriculum Study, Year 2: 2012-‐2013

Ben Satterfield, Ed.D.

The study of the effectiveness of the Equals Math Curriculum at the Developmental Learning Program in

Midlothian, Illinois was continued for a second year in the 2012-‐2013 school year. As in Year 1, instruction using

the Equals Math Curriculum was provided to participating students daily for forty (40) minute sessions in small

group (3 to 8 students per group) settings from September through May.

A total of 21 teachers and 88 students with disabilities in grades K-‐8 (ages 5-‐14) completed Equals instruction for

Year 2. Of this number, fifty-‐eight (58) were participating in the study for a second year. There were thirty (30)

students participating in the study for the first time. The disabilities of the participating students included

cognitive disabilities (mild, moderate, and severe), autism, multiple disabilities, emotional disorder, and other

health impairments.

Baseline information for all participants was collected in April, 2012. Post-‐intervention was assessed in April,

2013. All progress was recorded using the Equals assessment protocol.

Over the two year period, the participating students demonstrated an average gain of 56.74 as measured by the

Equals assessment protocol. This reflects average gains of 83% over the baseline recorded in September of 2011.

As in Year 1 of this study every student in the first cohort (Year 2) demonstrated progress. The average gain for

students participating in year two was 28.39. The average gain for these same students in the previous year was

28.35. Thus the average gain for students participating for a second year was slightly higher (0.04) than average

gain for the same students in the first year. Analysis of sub-‐groups in Year 2 indicates results similar to those

observed in Year 1.

All students in the second cohort (those entering the study for the first time) also demonstrated progress. The

average gain for this group was 27.02. These gains are comparable (1.37 points lower) to those of the first

cohort, indicating a similar rate of progress. Performance levels of subgroups in the second cohort were also

similar to those in the first group.

In Year 2 of the study, the collected data was expanded to include scores on the state standardized math test.

There are four scoring levels of this state math test: Mastery, Satisfactory, Foundational, and Entry. There were

45 students in the first cohort who took the state math test in both 2012 and 2013. Of these, eighty (80) percent

(36 out of 45) maintained or improved levels on state testing. Two additional students from the first cohort also

took the state math test – but only once. Of these 47 students, ninety-‐one (91) percent (43/47) students achieved

a Mastery or Satisfactory level on the state test. These are positive results for these students. The results from the second year of this study indicate steady progress for the students participating in this

prescribed instruction using the Equals Math curriculum. Many of the students in this study are among the most

challenging students to whom to teach math and achieve demonstrable progress. Students in this study have

demonstrated steady progress for two years. These students’ positive results on the state math testing lend

credibility to the findings of this study as measured using the Equals assessment protocol.

15

EQUALS MATH CURRICULUM

DLP STUDY – YEAR 2 DATA

Students Participating for 2 Years

Grouping N Sept

(2011)

April

(2012)

Mean

Difference Sep11-‐‐Apr12

p-‐Value

(0.05)

Δ YR 1

April

(2013)

Mean

Difference Apr13-‐Apr12

p-‐Value

(0.05)

Δ YR 2

Mean


p-‐Value

(0.05)

Δ YR 1&2

All Students 58 67.991 96.345 28.354 0.000 124.733 28.388 0.000 56.742 0.000

State Test Scores (N=57)

Advanced one performance level 6 Mastery Level 24

Maintained current performance level 30 Satisfactory Level 19

Dropped a level 9 Foundational Level 3

Insufficient data 13 Entry Level 1

Of students with sufficient data 80% (36 out of 45) maintained or improved math scores on state testing.

Disability N Sept

(2011)

April

(2012)

Mean


p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Autism

29 68.27 102.64 34.37 0.0000 129.71 27.07 0.0000 61.44 0.0000

Cognitive

Impairment

20 66.81 90.79 23.98 0.0000 120.18 29.39 0.0000 53.37 0.0000

Multiple

Disabilities

7 56.57 73.71 17.14 0.0013 103.43 29.72 0.0050 46.86 0.0009

Other Health

Impairment

2 82.25 105.75 23.50

149.25 43.50

67.00

Emotional

Disorder

1 194.50 225.5 31.00

240.00 14.50

45.50

16

Instructional

Level

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 17 26.09 48.12 22.03 0.0000 64.26 16.14 0.0000 38.17 0.0000

Level 2 27 67.13 91.24 24.11 0.0000 123.89 32.65 0.0000 56.76 0.0000

Level 3 14 120.54 164.75 44.21 0.0000 199.79 35.04 0.0000 79.25 0.0000

Gender N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Female 9 73.56 93.06 19.50 0.0000 120.00 26.94 0.0000 46.44 0.0000

Male 49 66.97 96.95 29.98 0.0000 125.60 28.65 0.0000 58.63 0.0000

Ethnicity N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Asian 1 54.50 78.50 24.00 99.50 21.00 45.00

African-‐‐

American

22 69.93 100.66 30.73 0.0000 129.02 28.36 0.0000 59.09 0.0000

Hispanic 13 61.45 98.85 37.40 0.0002 129.00 30.15 0.0000 67.55 0.0000

White 22 70.54 91.36 20.82 0.0000 119.07 27.70 0.0000 48.53 0.0000

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Socio-‐Economic

Status

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Free Lunch 41 68.39 99.17 30.78 0.0000 129.00 29.83 0.0000 60.61 0.0000

Reduced

Lunch

4 19.50 40.87 21.37 0.0190 55.25 14.38 0.0525 35.75 0.0154

Regular

Lunch

13 81.65 104.50 22.85 0.0002 132.65 28.15 0.0000 51.00 0.0000

Autism N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 6 17.83 47.91 30.08 0.0045 63.08 15.17 0.0208 45.25 0.0006

Level 2 16 65.09 92.15 27.06 0.0000 120.97 28.82 0.0000 55.88 0.0000

Level 3 7 110.07 164.00 53.93 0.0115 196.36 32.36 0.0041 86.29 0.0011

Cognitive

Disability

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 9 34.22 53.44 19.22 0.0000 71.33 17.89 0.0021 37.11 0.0001

Level 2 4 59.75 77.75 18.00 0.0509 114.75 37.00 0.1060 55.00 0.0197

Level 3 6 131.00 165.50 34.50 0.0000 198.11 32.61 0.0008 67.11 0.0007

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Multiple

Disability

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 2 14.25 24.75 10.50 __ 36.0 11.25 __ 21.75 __

Level 2 5 73.50 93.30 19.8 0.1780 130.40 37.1 0.0079 56.90 0.0007

Level 3 __ __ __ __ __ __ __ __ __ __

Other Health

Impairment

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 __ __ __ __ __ __ __ __ __ __

Level 2 2 82.25 105.75 23.50 149.25 43.50 __ 67 __

Level 3 __ __ __ __ __ __ __ __ __ __

Emotional

Behavior

Disorder

N Sept

(2011)

April

(2012)

Mean

Difference

p-‐‐

Value

(0.05)

April

(2013)

Mean


p-‐Value

(0.05)

Mean


p-‐Value

(0.05)

Δ YR 1&2

Level 1 __ __ __ __ __ __ __ __ __ __

Level 2 __ __ __ __ __ __ __ __ __ __

Level 3 1 194.50 225.50 31.00 __ 240 14.5 __ 45.50 __

19

Grouping N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

All Students

1st YR

30 40.5172 67.5345 27.0173 0.0000

Students Participating for 1st Year

State Test Scores (N=30)

Advanced one performance level 2 Mastery Level 4 Maintained current performance level 0 Satisfactory Level 2 Dropped a level 1 Foundational Level 1 Insufficient data 37 Entry Level 0

Of students with sufficient data 66% (2 out of 3) maintained or improved math scores on state testing.

Instructional Level N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Level 1 16 29.47 51.13 21.66 0.0000

Level 2 12 37.18 69.64 32.46 0.0000

Level 3 2 141.50 177.25 35.75

Disability N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 10 58.39 84.72 26.33 0.0031

Cognitive Disability 11 26.15 57.45 31.30 0.0000

Developmental Delay 5 45.25 72.00 26.7500 0.0008

20

Multiple Disability 3 19.25 31.50 12.25 0.3402

TBI 1 26.50 58.50 32.00

Level 1 N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 6 28.30 49.90 21.60 0.0810



Multiple Disability 3 19.25 31.50 12.25 0.3402

TBI 1 26.50 58.50 32.00


Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 3 40.50 79.00 38.50 0.2852



Multiple Disability 0

TBI 0

21


Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 1 262.00 284.00 22.00

Cognitive Disability 1 21.00 70.50 49.50

Developmental Delay 0

Multiple Disability 0

TBI 0

Ethnicity N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

African-‐American 14 28.96 55.00 26.04 0.0000

Hispanic 3 21.00 34.00 13.00 0.3512

White 11 57.75 82.55 24.80 0.0000

Other 2 13.25 49.5 36.25

Gender N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Female 7 36.12 58.42 22.30 0.0001

Male 23 42.25 69.27 27.02 0.0000

22

Socio Economic

Status

N April (2012)

Mean

April (2013)

Mean

Mean

Difference

p-‐Value (0.05)

Free Lunch 23 38.70 65.48 26.78 0.0000

Reduced Lunch 1 32.5 53.5 21.00

Regular Lunch 6 40.80 67.60 28.80 0.0144

23

Equals Math: A Framework of Math Instruction for Students with Disabilities Year 1: 2011-‐‐2012

Ben Satterfield, Ed.D. & Karen Ross-‐‐Brown, MLS

Overview The necessity for a mathematics curriculum that addresses the needs of students with disabilities has been identified. Equals Math is a K-‐‐12 standards-‐‐based accessible curriculum that employs features identified in research literature on math instruction for students with disabilities. A developmental learning program in a suburban Midwestern school district conducted a year-‐‐long test of the effectiveness of the Equals curriculum. There were 72 students (ages 5-‐‐14) who completed the study and 21 participating teachers. Data on student progress was collected from tests given pre-‐‐ and post-‐‐ instruction, using the assessment protocol that is included in Equals. Teacher feedback was gathered by a survey at the end of the year. All students demonstrated progress, including students with significant disabilities. Teachers reported that the Equals curriculum was effective in helping students develop math skills and that it provided a framework for teaching students who face many challenges in learning mathematics. Not only were teachers successful in providing access to math for all their students, but many students also exceeded their teachers' expectations in their performance. What Research Tells Us

Since the 1997 amendments to the Individuals with Disabilities Act (IDEA), there has been a pressing need for a math curriculum for students with disabilities that provide meaningful, standards-‐‐based math instruction (Warger, 2002; Council for Exceptional Children, 2011). Jimenez (2011) suggests that it is useful to see the need for such a curriculum in the context of changing curricular perspectives over the past four decades. In the 1970s students with significant disabilities of all ages were generally exposed to adaptations of early childhood mathematics curricula. In the 1980s math instruction for these students developed around functional life skills. In the 1990s emphasis was given to self-‐‐determination and social inclusion. In the 2000s access to general academic content was grafted into the prevailing trends set in motion earlier (social inclusion, functional skill development, self-‐‐determination).

Browder, Spooner, Ahlgrim-‐‐Delzell, Harris, & Wakeman (2008) conducted a meta-‐‐analysis of research on mathematics instruction for students with various disabilities. They found that the math instruction for students with significant disabilities consisted largely of computation, measurement, and money skills. Similarly, students with moderate intellectual disabilities have received math instruction focused around money skills, counting, number matching and identification, and telling time. These skills have most commonly been taught in the context of daily living activities such as shopping and following a schedule. While algebra had been part of the scope of instruction with some students with Learning Disabilities, Browder et al. found an

24

absence of investigations of instruction in standards-‐‐based mathematical skills for students with disabilities.

Jimenez (2011) identifies four areas of difficulty that students with significant disabilities often experience with standards based math instruction: (1) understanding basic math vocabulary (reading and discussion of math concepts); (2) familiarity with basic math operations (identifying and executing strategies for solving problems and equations); (3) exposure to math (lack of previous instruction and limited real-‐‐life experience with math); and (4) challenges with memory (remembering math facts and concepts).

Ahlgrim-‐‐Delzell, Knight, Jimenez, & Agnello (2009) identified a need for models of instruction for teachers of students with significant and moderate disabilities. They called for a clearly articulated conceptual framework for what it means to teach standards-‐‐based general academic content to these students. The dearth of information available about appropriate instruction for these students leaves teachers with little guidance with regard to effective instructional strategies.

Cawley (2002) found that presenting math concepts in a way that is meaningful helps students with disabilities develop understanding of how and when to apply these concepts. He suggests that much of math instruction emphasizes “doing” and neglects “knowing”. Gagnon & Maccini, (2001) found that students with disabilities come to math with poor computational and reasoning skills, and lack the ability to identify significant information in word problems. Nonetheless, with the proper supports in place, these students can experience success. Montague, Applegate, & Marquard (1993) contend that students with disabilities develop problem solving skill more effectively when they are taught how to think about problems (metacognitive processes) as well as how to “do” problems (cognitive strategies).

Maccini & Gagnon, (2000) assert that it is the deeper understanding of math that will support and serve students functionally in the real world, not surface knowledge or tricks. While it is important to identify portable, practical math tools for use in the community and at home (e.g., calculator), knowing when to use such a tool is a primary objective (Thompson & Sproule, 2000). In addition, when students are taught mathematics concepts in relationship to the system of numbers, they are better equipped to move forward with learning higher level thinking skills (Moore, 1973; Pellegrino and Goldman, 1987; Gersten, & Chard, 1999). Providing opportunities for students to work to their potential raises expectations. Higher expectations contribute to achievement (Englund, Luckner, Whaley, & Egeland, 2004; McKown, & Weinstein, 2008).

Often prerequisite skills have been embedded within a standard, as it was assumed students in general education already possessed those skills. However, students with disabilities often miss the experiences in and out of school that build those prerequisite skills due to limited opportunity, physical or cognitive access, medical needs, and/or communication deficits (Mercer, Jordan, & Miller, 1996; Mercer, Lane, Jordan, Allsopp, & Eisele, 1996; Mercer &

25

Mercer, 1998; Miller & Mercer, 1997). Thus proper sequencing of skills is necessary in order to address the learning needs of these students.

Students with disabilities typically need longer processing time, specialized materials, and adaptations to learn, apply, and practice math concepts; thus it often becomes necessary to break many objectives into much smaller chunks (Nadis, 1993; Miller & Koesling, 2009). It is more challenging for students with disabilities to learn multiple skills in a single lesson, as is common in general education curricula.

Equals Math Curriculum

AbleNet, Inc., an assistive technology and curriculum company for individuals with disabilities, accepted the challenge to develop a mathematics curriculum, Equals™ Mathematics. Equals was developed by drawing from general education math standards from around the country and best practice math instruction, learning strategies, and pedagogy from a wide variety of resources and research materials. These included NCTM recommendations and periodicals, publications from other sources (books, papers, journals, and periodicals) major math publishers, math instructors and special educators. In addition, action research was conducted in large and small school districts across the United States with K-‐‐12 students who have mild, moderate, and significant disabilities (Meyer, Ross-‐‐Brown, & Satterfield, 2010).

Equals instruction is organized around the five content standards identified by the National Council of Teachers of Mathematics (NCTM) which include: (1) Numbers and Operations, (2) Algebra, (3) Geometry, (4) Measurement, and (5) Data Analysis and Probability (NCTM, 2000). The curriculum incorporates the associated NCTM Process Standards which stress real-‐‐world connections, communication, multiple representations, reasoning, and problem solving in addition to knowledge of formulae and process.

Equals Mathematics was written to teach students math concepts and problem solving by using an array of tools and applying concepts to real life. The curriculum seeks to focus upon developing student understanding of when and how to use skills and concepts. Equals was designed to provide for students’ unique needs with clear, simple, and singular objectives. These objectives, taken from state math standards across all regions of the United States, were written into the program to provide a sequence of skills to include the necessary prerequisites and fit the pacing and learning needs of students with disabilities, including students with the most significant needs.

The length and breadth of the lessons in Equals have been crafted to provide meaningful, connected lesson objectives in a sequence that takes full advantage of the connectivity within and between the six math content areas identified within the curriculum (numbers and operations, geometry, algebra, data analysis and probability, measurement, and attending and exploring). Equals does not have grade level boundaries, so students can continue to learn from where they left off from year to year and school to school. However, as Equals was developed with grade level standards in mind, it represents a full math curriculum, addressing

26

skills typically taught in general education grades pre-‐‐K – 4 that are aligned to K-‐‐12 standards. The set of skills within this range are the skills students need to move forward successfully in mathematics and frequently the skills students with disabilities struggle to understand. The lessons are designed to be age-‐‐neutral so all students, no matter their age, feel they are using age-‐‐appropriate materials and can succeed.

The basic structure of each Equals lesson is consistent and systematic. Each lesson follows the same set of guiding principles with a consistent format and presentation. The components include the following best practice math instruction methodologies present in every lesson (directly related to each objective and written into the lesson in a progression of learning):

• activation of background knowledge by way of review of past related vocabulary and concept

• exploration for building background knowledge

• connection of new concept to background knowledge

• verbal communication about the math concept

• modeling written communication about the math concept

• concrete representation of math concept/vocabulary

• math vocabulary instruction • CSA (Concrete -‐‐ Semi-‐‐Concrete -‐‐ Abstract) learning sequence instruction and exploration

• skill practice

• problem solving instruction

• problem solving

• math journaling

• formative assessment

• sensory math activity

• real-‐‐life problem solving connections

• workstations

• games for practice

• home letter communication and support activities

Equals CSA Methodology

Equals derives its educational foundation from the Concrete, Semi-‐‐Concrete, Abstract (CSA) pedagogy found in general education mathematics curricula (Allsopp, 1999; Allsopp, et al., 2008; Fahsl, 2007; Jordon, Miller and Mercer, 1998). This instructional approach has been demonstrated to be successful with students in elementary grades (Allsopp, 1999; Allsopp, et al., 2008; Fahsl, 2007; Harris, Miller and Mercer, 1995, Jordan, et al., 1998). CSA further draws upon brain research showing that learning is retained longer when presented in multiple formats (Sousa, 2001). Jordan et al. (1998) demonstrated the efficacy of CSA instruction with students with disabilities. Nonetheless, CSA remains largely unused beyond early grades in many school settings (Bender, 2009).

27

The CSA approach supports teachers with the means to provide students the opportunity to use concrete objects, images, numbers, and math symbols to help them grasp math concepts. In this way the Equals curriculum helps each student learn math in the manner that they learn best and helps them build background knowledge and a wider view of all that math entails.

Fig. 1 Illustration of CSA

Equals Assessment

Teachers need assessments that are appropriate for learners and that provide information that can be used to identify where instruction may begin (Rulon, 2002). While traditional testing is convenient and effective for assessing the knowledge possessed by typical students, the process of assessing students with moderate and significant disabilities becomes much more difficult. Browder, Spooner, Ahlgrim-‐‐Delzell, Harris, & Wakeman (2008) found that teachers lacked a means of effectively assessing skills and the effectiveness of math instruction for these students in the five mathematical components identified by the NCTM (2000).

The Equals Math Curriculum has an assessment tool. This tool is based upon the Curriculum-‐‐ Based Measurement developed at the University of Minnesota (Deno, 1985). Designed to be administered to collect baseline data or to establish the entry point for instruction, the Equals assessment may be used as a pre-‐‐ and post-‐‐test to monitor and document progress. This tool is divided into six subtests that correspond to the NCTM components with the addition of one component that provides visibility to pre-‐‐emerging academic skills. The six Equals assessment sub-‐‐tests are: (1) Attending & Exploring, (2) Data Analysis and Probability, (3) Algebra, (4) Geometry, (5) Measurement, and (6) Numbers and Operations. When the Equals Assessment is administered, students are assessed in each subtest. Equals includes specific instructions to adapt test questions to help students demonstrate their knowledge. Students may respond to questions in a way that best meets their needs and abilities. Adaptations to test items and adapted responses are tracked and used to adjust the Equals raw score. A formula is employed to align the adjusted raw score with a suggested starting point for Equals instruction.

The primary task in developing the Equals assessment was to identify barriers to performance to understand better what students know and what they were thinking. These barriers were viewed as cognitive, motor, and language disabilities. The Equals assessment’s adapted test items are tasks with motor and expressive language performance removed, with additional

28

support given by way of visuals and manipulatives. Thus the tasks meet students where they are in their mathematical thinking while also staying true to the math concept being assessed. In this way, full and/or incremental math knowledge can be discovered and measured.

Differentiated Instruction and Increased Expectations

Differentiation is structured within each lesson in three levels. Teachers can choose a level to match a student's individual instructional needs and then change the level it as needed to address unique needs. While the initial choice of level is based on student needs, this provides a flexible system of differentiation from lesson to lesson.

Students with severe/profound disabilities have significant challenges with language, motor, and/or cognitive skills (Jimenez, 2011). Generally, these students require assistance throughout the day for safety, to complete daily cares, and for learning. Students with severe/profound disabilities typically benefit from instruction at Level 1.

Students with moderate disabilities have similar challenges as students with severe disabilities but to a lesser degree, although a student with moderate disabilities can have significant challenges with motor and/or language skills. When supports are placed in the environment, students with moderate disabilities can usually perform independently. Students with moderate disabilities benefit from instruction at Level 2.

Students with mild disabilities have fewer challenges with cognitive skills. Language and/or motor disabilities may be present to a significant degree, but not across all three areas. Students with mild disabilities benefit from instruction at Level 3.

While teachers look to the three levels to differentiate instruction, they are meant to be flexible, as unique needs vary greatly from student to student. The levels are meant to be used as a guide for teachers to choose and adjust as they see fit.

With three levels embedded in every lesson, worksheet, workstation, and test, assumptions of ability for individual students are eliminated and expectations are raised. The principle applied here is that all of math is for every student.

Fig. 2 Example of Equals’ leveled approach to differentiation

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Math Action Dictionary Differentiation Guide

In addition to the three levels, a Math Action Dictionary is provided to address access needs beyond specifics within lessons. This is a teacher resource for adaption, assistive technology, and differentiation ideas. This tool provides a global resource for differentiation based on, and organized by, actions students are asked to perform in the lessons. When a student is unable to perform an action in a typical way, the teacher can locate alternative ways in the Math Action Dictionary. Each action word entry includes 3-‐‐4 technology-‐‐based and no-‐‐tech solutions to increase engagement and ensure all students are active learners.

Research Methodology

While preliminary action research indicated that Equals was an effective Math curriculum for students with disabilities (Meyer, Ross-‐‐Brown, & Satterfield, 2010), no systematic study had been attempted. A study of the effectiveness of the Equals Mathematics curriculum was undertaken by the staff at the Developmental Learning Program from a suburban special education school district in the mid-‐‐western United States. A One-‐‐Group Pre-‐‐test Post-‐‐test research design was selected (Campbell & Stanley, 1963). The goal was to record student progress over the course of one school year. Teachers were asked to report the positive and negative aspects of the curriculum they encountered. A total of 21 teachers and 107 students with disabilities in grades K-‐‐8 (ages 5-‐‐14) were invited to take part. Of this number, data was collected on 72 students who participated for the entire year. The disabilities of the participating students included cognitive disabilities (mild, moderate, and severe), autism, multiple disabilities, and other health impairments.

Instruction in the Equals Math Curriculum was provided to participating students daily for forty (40) minute sessions in small group (3 to 8 students per group) settings. Instruction continued daily from September through May of the same school year. Baseline data was collected in September, at the outset of the school year, before the commencement of Equals instruction. Progress was assessed in April. Baseline and end-‐‐of-‐‐year progress was recorded using the Equals assessment protocol. Both pre and post assessment data was collected by staff trained in the Equals assessment protocol, but who were unaffiliated with the students they assessed.

Teachers involved in the project participated in two two-‐‐hour training workshops on the Equals curriculum provided by AbleNet. The assessment team who would administer the pre and post testing also received training in Equals with attention to administration of the assessment itself.

30

Assessment

The Equals assessment was administered to the students in this study in September of 2011, at the outset of the school year, to determine each student’s entry point for Equals instruction. This data also served as baseline data for this study. Students were assigned ID numbers for tracking purposes. Adjusted raw scores were associated with these ID numbers for later reference. After eight months of Equals instruction, the Equals assessment was administered again. Adapted raw scores were collected for each student ID and compared. Progress was measured by the difference in Adjusted Raw Scores between September and April.

The Equals assessment was administered by a team of educators from the school that had been trained in the Equals assessment protocol. However, these individuals did not conduct the Equals instruction for students they taught during the year. This was done to assure impartiality in the data collection process.

In addition, teachers were given a 12-‐‐question Likert survey (see Appendix 2) asking them to rate the effectiveness and ease of use of the Equals curriculum. Opportunity to make open-‐‐ ended comment was provided teachers at the end of the survey questions.

Analysis

At the end of the study, individual student progress, as measured by the Equals Assessment protocol, was examined and analyzed. Calculation of actual progress for each student was made by subtracting the September adjusted raw score from the April adjusted raw score. Analysis of the progress was conducted by the instructional level in Equals, disability, age, grade, ethnicity, gender, and socio-‐‐economic status.

Analysis of Student Results

The result of the baseline (September) adjusted raw score for each student was compared to the adjusted raw scores when the assessment was administered the following April. Progress was measured by the difference in the two scores (see Appendix 1).

This analysis indicates that every one of the 72 students that completed the year demonstrated progress. No student regressed. Taken together the mean difference for all students was 28.04. This represents an average gain in math skills of 38%. Certain subgroups demonstrated even greater gains.

Students demonstrated progress at all three instructional levels. Students participating at Level 1 progressed by an average of 21.86. Level 2 students produced average gains of 24.05. Those engaged in Level 3 instruction demonstrated an average gain of 44.44.

31

Students made progress regardless of disability. Students with a diagnosis of autism reflected a mean difference of 32.15. Results also demonstrated noteworthy progress for students with cognitive disabilities (25.27) and multiple disabilities (21.21).

When disability group was examined in the light of instructional level, progress was evident at every level across all disabilities. Students with autism who were participating in Equals instruction at Level 1 demonstrated impressive gains (30.09). Level 2 students with autism progressed, reflecting a mean difference of 24.17. Those with autism in Level 3 instruction posted even greater progress (52.16).

Among students with cognitive disabilities, those engaged in Level 3 instruction produced average gains of 35.08. However, progress was also evident for students with cognitive disabilities participating at Level 1 (19.20) and Level 2 (25.50). Students with multiple disabilities also demonstrated progress at Level 1 (10.50) and Level 2 (22.22). No students with multiple disabilities were engaged in Level 3 instruction. Two students with other health impairments participated, demonstrating progress as well. One student was at Level 2 (23.50) and the other was at Level 3 (31.00).

When ethnicity was examined it was evident that students from all backgrounds made progress with Equals. Hispanic (34.88) and African-‐‐American (31.62) students demonstrated notable progress under Equals instruction. White (19.40) and Asian (24.00) students also showed gains. When ethnicity is examined in conjunction with instructional level, African-‐‐American students demonstrated greater progress in both Level 2 (29.56) and Level 3 (47.17) instruction than the average of all students. African American students at Level 1 evidenced progress as well (21.00). Hispanic students also produced higher gains in Level 1 (30.10) and Level 3 instruction (53.60) than those of all students taken together. Hispanic students in Level 2 made progress (24.93) as well. White students in Level 1 (16.33) and Level 2 (15.96) demonstrated progress, but White students in Level 3 showed above average progress (32.00).

Socio-‐‐economic status (as indicated by a student’s participation in the free-‐‐lunch program) also produced marked differences. Participants in the free-‐‐lunch program presented a difference of 30.09, while their reduced lunch counterparts demonstrated a difference of 21.75, and 24.67 for students on the regular lunch program. When viewed in relation to instructional level, students with free lunch involved in level 3 instruction produced gains of 52.1.

The analysis indicated that both boys (29.42) and girls (19.45) made progress, as did students in each of the grade levels examined. The most prominent gains were demonstrated by second grade students (37.50) and seventh graders (29.46).

Test for Statistical Significance

To ensure that these results were not the result of random factors, a test for statistical significance of this data was conducted by applying a two tailed t-‐‐test for two paired sample for

32

means at the 0.05 level. This calculation produces a result known as “p-‐‐value”. When the p-‐‐ value calculation is less than 0.05, we can have confidence that the results are not random. This analysis found p=0.00 for all students and p<0.05 for all sub groups with the exception of sub-‐‐ groups with very few participants (see Appendix 1).

Analysis of Teacher Surveys

Twenty-‐‐one teachers took part in a survey at the conclusion of the school year to assess the effectiveness of the Equals Math curriculum used with their students. Twelve questions were asked in a five-‐‐point Likert scale survey (where 1 is least effective and 5 is most effective). Opportunity to comment was then provided (see Appendix 2).

Descriptive statistics (mean and median calculations) were used to analyze the results of the survey. Ten of the twelve items received mean scores of 4.0 or higher with medians of 4 or higher. The highest rating (4.62) and greatest consensus among the teachers (all teachers gave this item a 4 or 5) related to how effective the Equals curriculum’s spiraling design was for students.

Equals also received high scores on questions relating to increasing effectiveness at producing student learning across a range of math concepts (4.57), and regarding the effectiveness of the curriculum at developing content and skills (4.52). Both questions reflected strong levels of consensus and posted median scores of 5.

Several other aspects of the Equals curriculum also drew high marks: the effectiveness of the manipulatives associated with Equals (4.48), the effectiveness of the Equals assessment for placement purposes (4.39) the effectiveness of the curriculum’s organization (4.38), the effectiveness of the layout of Equals lesson for lesson planning (4.33) and the effectiveness of Equals for group instruction as opposed to previous group instructional strategies (4.33). Lesson layout and chapter evaluation effectiveness each received ratings of 4.00.

From the survey results, it is clear that the participating teachers regarded the Equals curriculum as well-‐‐organized and effective. They affirmed that Equals provided a framework within which to teach math to their students. Teachers confirmed that the use of Equals produced learning beyond what had been previously expected of their students.

Teacher Comments and Observations

Teacher comments echoed the survey results. Several teachers remarked that the Equals curriculum simplified lesson planning and made them feel more productive. One teacher noted: “The planning is very simplified. It is awesome to have a curriculum that is already modified for so many levels.”

Other teachers commented upon the success experienced by their students. “The curriculum has allowed our students the opportunity to learn important concepts and skills which they

33

wouldn’t necessarily be exposed to… The student growth over the past [school year] coupled with the students ability to articulate math concepts has been outstanding.”

Another teacher commented: “It’s wonderful to see my students’ use of the math vocabulary when they aren’t in math class.”

The majority of teacher comments, however, indicated that the systematic organization of the Equals curriculum was a major factor. “The organization of the teacher’s guide, lessons and materials is very teacher-‐‐friendly. It really makes it easy for the teachers to teach and reinforce concepts.”

One teacher noted: “the curriculum presents concepts in a manner where ideas are presented and built upon in a way which seems to make sense to our students.” Another echoed her appreciation for “the sequencing of skills that continue to build throughout the program.”

Regarding the ability of Equals to support multiple learning styles, one teacher stated: “The equals math curriculum has a variety of interactive activities included in every lesson, making it easy to reach different types of learners.”

A teacher stated that Equals was “…well organized. [There are] multiple levels and approaches to teach a variety of learners. Data driven programming means a much less subjective approach to teaching math to my students.”

Summary

It is clear from the results of the Equals assessment that all students in this study demonstrated progress in the Equals Math curriculum. Survey responses, comments, and observations from the teachers in this project confirm these findings.

As students on all three instructional levels demonstrated progress, the results appear to support the Equals strategy of presenting three levels of differentiation and the CSA structure within each lesson. Again, teacher survey results and comments affirm the effectiveness of this approach.

The fact that students engaged in Level 1 and Level 2 instruction made progress in the Equals curriculum in eight months is itself remarkable. The students in these levels are generally those with moderate to severe intellectual disabilities and those our schools have struggled to connect to the general education curriculum. They are often given alternative assessments instead of standardized testing in the school setting. To have a systematic math curriculum that can support effective direct instruction with measureable outcomes for these students would be a tremendous advantage.

34

It is also noteworthy that certain sub-‐‐groups appear to have made remarkable gains in this study. Students with autism presented elevated levels of progress. Results indicate that African–American and Hispanic as well as those from lower socio-‐‐economic backgrounds demonstrated greater gains (based upon the Equals assessment) than the average of all students taken together.

This is especially meaningful because students with disabilities from minority families and those from lower income areas often have the least exposure to real-‐‐life math. These students often have gaps in their math knowledge and lack the breadth and quality of experience as their same-‐‐age peers without disabilities. Students with autism also have challenges learning mathematics, as it is problematic for teachers to identify the gaps and limits of their understanding. It appears that the students in this study from these sub-‐‐groups, were particularly successful in the Equals curriculum and were able to demonstrate remarkable gains in math skill.

Teacher surveys and comments suggested that the structure of the Equals curriculum was simple for teachers to follow, well organized, and effective with students. The ease of use appeared to give teachers confidence. Teachers remarked that the structure provided by Equals gave them a framework for addressing skill and concept development for students who had formerly been challenging to teach mathematics.

Future Research

This study is the first to take a pre-‐‐experimental or quasi-‐‐experimental approach to examine the Equals Math curriculum. It will be instructive to see how the results of future studies might expand upon these findings. Of particular interest will be the relative progress demonstrated by Level 1 and Level 2 participants by other various sub-‐‐groups. Longer term studies of progress by students who have learned math within the Equals framework for multiple school years may also yield valuable insights.

While the Equals Assessment has been developed based upon the Curriculum-‐‐Based Measurement developed at the University of Minnesota (Deno, 1985), it would be instructive to compare the progress demonstrated by students engaged in the Equals curriculum to progress in other, more standardized frameworks. This will be challenging, given the fact that traditional testing does not provide a standardized format that is accessible for students with disabilities and that can measure incremental knowledge.

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References

Ahlgrim-‐‐Delzell, L., Knight, V., Jimenez, B., & Agnello, B. (2009). Research based practices for creating access to the general curriculum in science for students with significant intellectual disabilities [Monograph]. Assessing Special Education Students, State Collaborative on Assessment and Student Standards, 1-‐‐41.

Allsopp, D. (1999). Using modeling, manipulatives, and mnemonics with eighth grade students. Teaching Exceptional Children, 32(2), 74–81.

Allsopp, D., Kyger, M., Lovin, L., Gerretson, H., Carson, K., & Ray, S. (2008). Mathematics dynamic assessment, Informal assessment that responds to the needs of struggling learners in mathematics. Teaching Exceptional Children 40(3), 6-‐‐16.

Allsopp, D., Lovin, L., Green, G., & Savage-‐‐Davis, E. (2003). Why Students with Special Needs Have Difficulty Learning Mathematics and What Teachers Can Do to Help. Teaching Mathematics in the Middle School, (8), 6, pp. 308-‐‐314.

Bender, W. N. (2009). Differentiating math instruction: strategies that work for K-‐‐8 classrooms. 2nd ed. Thousand Oaks, Calif.: Corwin.

Browder, D. M., Spooner, F., Ahlgrim-‐‐Delzell, L., Harris, A. & Wakeman, S. Y. (2008). A Meta-‐‐ analysis for teaching mathematics to individuals with significant cognitive disabilities. Exceptional Children, 74(4), 404-‐‐432.

Campbell, D. T., & Stanley, J. C. (1963). Experimental and quasi-‐‐experimental designs for research on teaching. In N. L. Gage (Ed.), Handbook of research on teaching (pp. 171–246). Chicago, IL: Rand McNally.

Campbell, D. T., & Stanley, J. C. (1966). Experimental and quasi-experimental designs for research. Boston: Houghton Mifflin Company.

Cawley, J. (2002). Mathematics interventions and students with high incidence disabilities. Remedial and Special Education, 23(1), 2-‐‐6.

Council for Exceptional Children (2011). Common Core Standards: What Special Educators Need to Know. Retrieved October 19, 2012 from: http://www.cec.sped.org/AM/Template.cfm?Section=CEC_Today1&TEMPLATE=/CM/ContentDi splay.cfm&CONTENTID=15269

Deno, S.L. (1985). Curriculum-‐‐based measurement: The emerging alternative. Exceptional Children, 52(3), 219–32

36

Englund, M. M., Luckner, A. E., Whaley, G. J., & Egeland, B. (2004). Children's Achievement in Early Elementary School: Longitudinal Effects of Parental Involvement, Expectations, and Quality of Assistance. Journal of Educational Psychology, 96(4), 723.

Fahsl, A. J. (2007). Mathematics accommodations for all students. Intervention in School and Clinic, 42(4), 198-‐‐203.

Gagnon, J., & Maccini, P. (2001). Preparing students with disabilities for algebra. Teaching Exceptional Children, 34(1), 8-‐‐15.

Gersten, R. & Chard, D. J. (1999). Number Sense: Rethinking Arithmetic Instruction for Students with Mathematical Disabilities. Journal of Special Education , 33 (1) 18-‐‐28. (Retrieved Nov. 28, 2012 from: http://enumeracy.com/files/Students%20with%20Mathematical%20Disabilities.pdf)

Harris, C.A., Miller, S.P., & Mercer, C.D. (1995). Teaching initial multiplication skills to students with disabilities in general education classrooms. Learning Disabilities Research & Practice, 10, 180–195.

Jimenez, B. (March, 2011) Teaching Grade Aligned Math & Science to Students with Significant Intellectual Disabilities. Presentation at Delaware Inclusion Conference. Retrieved December 28, 2012 from: http://www.doe.k12.de.us/infosuites/students_family/specialed/New/files/March2011.DE.Incl. Conf.pdf

Jordan, L., Miller, M., & Mercer, C. D. (1998). The effects of concrete to semi-‐‐concrete to abstract instruction in the acquisition and retention of fraction concepts and skills. Learning Disabilities: A Multidisciplinary Journal, v9 n3 p115–122 Fall 1998-‐‐Win 1999

Maccini, P., & Gagnon, J. C. (2000). Best practices for teaching mathematics to secondary students with special needs. Focus on Exceptional Children, 32(5).

McKown, C., & Weinstein, R. S. (2008). Teacher expectations, classroom context, and the achievement gap. Journal of school psychology, 46(3), 235-‐‐261.

Meyer, A. Ross-‐‐Brown, K, & Satterfield, B.(2010). Equals Mathematics White Paper. Retrieved October 5, 2012 from: http://www.ablenetinc.com/Portals/0/PDFs/EqualsWhitepaper.pdf

Mercer,C. D., Jordan L, Miller, S. P. (1996). Constructivistic math instruction for diverse learners. Learning Disabilities Research & Practice, 11, 147-‐‐156.

http://enumeracy.com/files/Students%20with%20Mathematical%20Disabilities.pdf)

http://www.doe.k12.de.us/infosuites/students_family/specialed/New/files/March2011.DE.Incl

http://www.doe.k12.de.us/infosuites/students_family/specialed/New/files/March2011.DE.Incl

http://www.ablenetinc.com/Portals/0/PDFs/EqualsWhitepaper.pdf

37

Mercer, C. D., Lane, H. B., Jordan, L, Allsopp, D. H., & Eisele, M. R. (1996). Empowering teachers and students with instructional choices in inclusive settings. Remedial and Special Education, 17, 226-‐‐236.

Mercer, C. D. & Mercer, A. R. (1998). Teaching students with learning problems (5th ed.). Upper Saddle River, NJ: Prentice-‐‐Hall Inc

Miller, P., & Koesling, D. (2009). Mathematics Teaching for Understanding Reasoning, Reading and Formative Assessment. The Right to Literacy in Secondary Schools: Creating a Culture of Thinking. New York: Teachers College, Columbia University, 65-‐‐79.

Miller, S.P. & Mercer, C.D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30 (1), 47-‐‐56.

Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy instruction and mathematical problem-‐‐solving performance of students with learning disabilities. Learning Disabilities Research and Practice, 8(4), 223-‐‐232.

Moore, M. (1973). Development of number concept in blind children. Education of the Visually Handicapped, V, 65-‐‐71.

Nadis, S. “Kid’s Brainpower: Use It or Lose It.” Technology Review, November/December 1993, 19–20.

NCTM (2000). Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA.

Pellegrino. J. W.. & Goldman. S. R. (1987). Information processing and elementary mathematics. Journal of Learning Disabilities, 20, 23-‐‐32, 57

Rulon, Michael (2002). Authenticity: The key to standards-‐‐based assessment. Classroom Leadership.

Sousa, D. A. (2001). How the Brain Learns: A Classroom Teacher’sGuide (2nd ed.). Thousand Oaks, California. Corwin Press.

Thompson, A., & Sproule, S. (2000). Deciding when to use calculators. Mathematics Teaching in the Middle Schoo1, 6(2), 126-‐‐ 129.

Warger, C. (July, 2002) Helping Students with Disabilities Participate in Standards-‐‐Based Mathematics Curriculum. Council for Exceptional Children. Accessed 11/16/2012 from: http://www.cec.sped.org/AM/Template.cfm?Section=Home&ContentID=1796&template=/CM/ ContentDisplay.cfm

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Appendix 1

EQUALS Math Curriculum

Pre & Post Testing

Grouping N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

All Students 72 73.42 101.46 28.04 0.000

Disabilities N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 36 76.75 108.90 32.15 0.000

Cognitive Disabilities 24 70.17 95.44 25.27 0.000

Multiple Disabilities 12 69.96 91.17 21.21 0.000

Instructional Level N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Level 1 18 25.33 47.19 21.86 0.000

Level 2 38 76.16 100.21 24.05 0.000

Level 3 16 121.03 165.99 44.96 0.000

Grade Level N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Grade K-‐1 9 53.28 80.39 27.11 0.001

Grade 2 7 45.79 83.29 37.50 0.006

Grade 3 10 66.15 95.10 28.95 0.001

Grade 4 15 57.20 82.97 25.77 0.000

Grade 5 9 79.72 108.22 28.50 0.000

Grade 6 3 145.00 166.00 21.00 0.052

Grade 7 12 90.54 120.00 29.46 0.002

Grade 8 7 104.00 127.29 23.29 0.028

Race N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

African-‐American 29 78.91 110.53 31.62 0.000

Hispanic 17 59.71 94.59 34.88 0.000

White 24 79.83 99.23 19.40 0.000

Asian 1 59.50 78.50 24.00 0.000

Other 1 12.50 31.50 19.00 0.000

Gender N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Girls 10 67.45 86.90 19.45 0.000

Boys 62 74.39 103.81 29.42 0.000

Socio-‐Economic

Status

N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Free Lunch Program 49 71.22 101.31 30.09 0.000

Reduced Lunch 8 59.94 81.69 21.75 0.0003

Regular Lunch 15 87.80 112.47 24.67 0.0002

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EQUALS Math Curriculum

Pre & Post Testing

Level by Disabilities by Socio-‐Economic Status

Grouping N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

All Students 72 73.42 101.46 28.04 0.0000

Instructional Level N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Level 1 18 25.33 47.19 21.86 0.0000

Level 2 38 76.16 100.21 24.05 0.0000

Level 3 16 121.03 165.99 44.44 0.0000

Disabilities N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 36 76.75 108.90 32.15 0.000



Level 1 N Sept (2011)

Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 6 17.83 47.92 30.09 0.0009




Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 21 77.93 102.10 24.17 0.0000



Other 2 82.25 105.75 23.50 0.1718


Mean

April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Autism 9 113.28 165.44 52.16 0.0008


Other 1 194.50 225.50 31.00 NA

40

Level 1

N

Sept (2011) Mean April (2012)

Mean

Mean

Difference

p-‐Value (0.05)

Free 11 26.64 48.68 22.05 0.000237

Reduced 4 15.75 38.75 23.00 0.008990

Regular 3 33.33 53 19.67 0.066135

Level 2

N


Mean

Mean

Difference

p-‐Value (0.05)

Free 28 71.95 97.34 25.39 0.000000

Reduced 3 82.83 101.83 19 0.149585

Regular 7 90.14 111 20.86 0.029386

Level 3

N


Mean

Mean

Difference

p-‐Value (0.05)

Free 10 118.25 170.35 52.1 0.000071

Reduced 1 168 193 25 NA

Regular 5 117.2 150.2 33 0.029864

Level 1

N


Mean

Mean

Difference

p-‐Value (0.05)

African-‐Am. 6 34.33 55.33 21.00 0.008836

Hispanic 5 17.4 47.5 30.10 0.003541

White 6 25.08 41.42 16.33 0.003734

Other 1 12.50 31.50 19.00 NA

Level 2

N


Mean

Mean

Difference

p-‐Value (0.05)

African-‐Am. 18 79.03 108.58 29.56 0.000013

Hispanic 7 61.93 86.86 24.93 0.000042

White 13 79.85 95.81 15.96 0.000570

Level 3

N


Mean

Mean

Difference

p-‐Value (0.05)

African-‐Am. 6 119.08 166.25 47.17 0.008598

Hispanic 5 98.9 152.5 53.60 0.014073

White 5 145.5 177.5 32.00 0.002076

41

Appendix 2

Developmental Learning Program Survey

1. How effective is the lesson layout with regards to lesson planning?

Least Effective Most Effective

1 2 3 4 5

2. How effective is the lesson layout with regards to execution of the lesson?


1 2 3 4 5

3. How effective is the curriculum organization?


1 2 3 4 5

4. How effective is the curriculum content/ skills taught with regards to student growth?


1 2 3 4 5

5. How effective is the spiraling curriculum (continually building upon skills in later lessons) to our

students?


1 2 3 4 5

6. How effective has the Equals program been at increasing student knowledge of a variety of math

concepts?


1 2 3 4 5

7. How effective has the planning and implementation of lessons in a group setting been in contrast to

previous practice (individual lessons for each student)?


1 2 3 4 5

8. How effective are the skill drill worksheets?


1 2 3 4 5

9. How effective are the problem solving worksheets?


1 2 3 4 5

10. How effective are the chapter assessments?


1 2 3 4 5

11. How effective are the manipulatives in conjunction with the lesson?


1 2 3 4 5

12. How effective is the students’ placement test in correctly assessing students for placement?

Least Effective Most Effective 1 2 3 4 5

Please share below any comments you might have:

42

Appendix 3

Developmental Learning Program Survey

Results

No. Question Mean Median

1. How effective is the lesson layout with regards to lesson

planning?

4.33 4

2. How effective is the lesson layout with regards to execution

of the lesson?

4.00

4

3. How effective is the curriculum organization? 4.38 4

4. How effective is the curriculum content/skills taught with

regards to student growth?

4.52

5

5. How effective is the spiraling curriculum (continually building

upon skills in later lessons) to our students?

4.62

5

6. How effective has the Equals program been at increasing

student knowledge of a variety of math concepts?

4.57

5

7. How effective has the planning and implementation of

lessons in a group setting been in contrast to previous

practice (individual lessons for each student)?

4.33

4

8. How effective are the skill drill worksheets? 3.76 4

9. How effective are the problem solving worksheets? 3.33 3

10. How effective are the chapter assessments? 4.04 4

11. How effective are the manipulatives in conjunction with the

lesson?

4.48

5

12. How effective is the students’ placement test in correctly

assessing students for placement?

4.39

4

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