FAST: earlyMath
Composite
Rating Summary
Classification Accuracy  

Generalizability  Moderate High  
Reliability  
Validity  
Disaggregated Reliability and Validity Data  
Efficiency  
Administration  Individual  
Administration & Scoring Time  58 Minutes  
Scoring Key  Computer Scored  
Benchmarks / Norms  Yes 
Cost 
Technology, Human Resources, and Accommodations for Special Needs 
Service and Support 
Purpose and Other Implementation Information 
Usage and Reporting 
The Composite measure is part of earlyMath in the FAST assessment suite from FastBridge Learning, LLC. earlyMath is designed to screen and monitor early academic readiness and achievement for students in Kindergarten and 1^{st} grade. The Formative Assessment System for Teachers (FAST™) is a cloudbased suite of assessment and reporting tools for reading, mathematics, behavior, which requires no additional costs for hardware or additional materials. FAST, developed by researchers at the University of Minnesota, is supported by an extensive set of materials to support teachers and students, including selfdirected training modules that allow teachers to become certified to administer each of the FAST assessments. The entire FAST assessment package, including online certification training and support, is provided at an annual rate of $6 per student.

Testers will require less than one hour of training. Paraprofessionals can administer the test. 
Where to Obtain: www.fastbridge.org Address:
FastBridge Learning
520 Nicollet Mall, Suite 910
Minneapolis, MN 554021057
Phone: 6122542534 Website: www.fastbridge.org Training materials are included in the cost of the tool. Additional, optional onsite and webinarbased training services are available for a fee. Ongoing technical support is available by calling 6124243714 or emailing help@fastbridge.org 
The earlyMath Composite measure of in the FAST assessment suite assesses both unified and component skills associated with Kindergarten and 1^{st} grade mathematics achievement. earlyMath is intended to enable screening and progress monitoring across three domains of mathematics (Number, Relations and Operations) and provide domain specific assessments of these component skills and a general estimate of overall mathematics achievement. The composites for Kindergarten and 1^{st} grade consist of 3–4 subtest measures (both timed and untimed). The development of earlyMath is based on a thorough examination of the most recent research literature and professional consultation in test development and mathematics education. Each of the subtests is aligned with National Common Core State Standards (CCSS, 2010) and three domains of number sense: (a) number, (b) relations, and (c) operations (Purpura & Lonigan, 2013; National Research Council, 2009). 
earlyMath is an assessment suite used to screen students in early numeracy skills in Kindergarten and 1^{st} grade. The Composite measure is intended for use in K–1 or with ages 5–7. The assessment is individually administered in 5–7 minutes per student. Scoring requires an additional one minute. Available scores include: raw scores, percentile scores, developmental benchmarks and cut points, subscales, composite, and error analysis. Scoring key is provided in the Administration Materials. 
Classification Accuracy
Classification Accuracy in Predicting Proficiency on GRADE 


Kindergarten 15^{th} Percentile n = 105 
1^{st} Grade 15^{th} Percentile n = 83 
False Positive Rate 
0.20 
0.25 
False Negative Rate 
0.18 
0.25 
Sensitivity 
0.82 
0.75 
Specificity 
0.81 
0.75 
Positive Predictive Power 
0.33 
0.13 
Negative Predictive Power 
0.97 
0.98 
Overall Classification Rate 
0.81 
0.75 
AUC (ROC) 
0.89 
0.93 
Base Rate 
0.10 
0.15 
Cut Points: 
54 
21 
At XX% Sensitivity, Specificity equals 
91% sensitivity, 0.68 specificity 
89% sensitivity, 0.57 specificity 
At XX% Sensitivity, Specificity equals 
82% sensitivity, 0.83 specificity 
81% sensitivity, 0.82 specificity 
At XX% Sensitivity, Specificity equals 
73% sensitivity, 0.85 specificity 
69% sensitivity, 0.85 specificity 
Generalizability
Description of Kindergarten study sample:
· Number of States: 12 (CO, IA, IL, IN, MA, MT, VT, OR, PA, WI)
· Regions: Midwest, Northeast, West
· Gender
o 51% Male
o 48% Female
· Race/Ethnicity:
o 52% White, NonHispanic
o 1% American Indian/Alaska Native
o 3% Black, NonHispanic
o 1% Asian, Pacific Islander
o 7% Hispanic
o 34% Unknown
Description of 1^{st} Grade study sample:
· Number of States: 6 (IA, IN, MA, MN, NY, VT)
· Regions: Midwest & Northeast
· Gender
o 49% Male
o 51% Female
· Eligible for free or reducedprice lunch: 32%
· Race/Ethnicity:
o 52% White, NonHispanic
o 1% American Indian/Alaska Native
o 3% Black, NonHispanic
o 1% Asian, Pacific Islander
o 7% Hispanic
o 34% Unknown
Reliability
Type of Reliability 
Age or Grade 
n (range) 
Coefficient 
SEM 
Information (including normative data)/Subjects 

range 
median 

Testretest 
K 
37 

.87 

10% Black, 8% Hispanic, 82% White, 15% Free and reduced lunch, 5% IEP eligible. See Appendix B for additional background information. 
Testretest 
1 
30 

.91 

3% American Indian, 13% Asian, 21% Black, 5% Hispanic, 59% White, 38% Free and Reduced Lunch, 3% IEP eligible. See Appendix B for additional background information. 
Coefficient Alpha  K  615  .96  Subsample of larger subject pool. See Appendix B for additional background information.  
Coefficient Alpha  1  58  .88  Subsample of larger subject pool. See Appendix B for additional background information. 
Validity
Type of Validity 
Age or Grade 
Test or Criterion 
n (range) 
Coefficient (if applicable) 
Information (including normative data)/Subjects 

range 
Median 

Concurrent 
1 
Measures of Academic Progress for Primary Grades – Math (MAP) 
200 

0.65 
Data collected in Fall. 
Concurrent 
K 
MAP 
223 

0.71 
Data collected in Winter. 
Concurrent 
1 
MAP 
187 

0.65 
Data collected in Winter. 
Concurrent 
K 
MAP 
225 

0.57 
Data collected in Spring 
Concurrent 
1 
MAP 
175 

0.59 
Data collected in Spring 
Predictive 
K 
MAP 
212 

0.66 
Fall to Spring prediction. 
Predictive 
1 
MAP 
192 

0.68 
Fall to Spring prediction. 
Predictive 
K 
MAP 
221 

0.69 
Winter to Spring prediction. 
Predictive 
1 
MAP 
186 

0.66 
Winter to Spring prediction. 
Predictive 
K 
MAP 
214 

0.72 
Fall to Winter prediction. 
Predictive 
1 
MAP 
193 

0.69 
Fall to Winter prediction. 
Predictive 
K 
GMADE composite Level R 
140 

0.56 
Fall to Spring prediction. 
Predictive 
1 
GMADE composite Level 1 
155 

0.67 
Fall to Spring prediction. 
Predictive 
K 
GMADE composite Level R 
143 

0.56 
Winter to Spring prediction. 
Predictive 
1 
GMADE composite Level 1 
158 

0.68 
Winter to Spring prediction. 
Concurrent 
K 
GMADE composite Level R 
152 

0.63 
Data collected in Spring 
Concurrent 
1 
GMADE composite Level 1 
153 

0.69 
Data collected in Spring 
Predictive  1  MAP for Primary Grades  Math Number Sense Strand  192  0.71 
Fall to Spring Prediction Students from one district in Minnesota. 3% American Indian, 7% Asian, 3% Black, 4% Hispanic, 84% White. 50% Female, 7% eligible for special education services, 35% eligible for free and reduced lunch 

Predictive  1  MAP for Primary Grades  Math Computation Strand  192  0.71 
Fall to Spring Prediction See above for demographics. 

Concurrent  K  aMath  191  0.76 
End of Year Students from 2 schools in Minnesota 53% Female, 73% Race Not Specified 

Predictive  K  aMath  185  0.82 
Early Spring to Late Spring Students from 9 schools in Minnesota. 51% Female, 85% Race not Specified 

Predictive  K  aMath  245  0.76 
Early Spring to End of Year Students from 9 schools in Minnesota. 52% Female, 30% White, 69% Race not Specified 

Concurrent  1  aMath  1616  0.70 
Winter Students from 35 schools in Iowa, Indiana, Massachusetts, Minnesota, New York, and Vermont. 1% American Indian, 2% Asian, 5% Black, 9% Hispanic, 59% White, 27% not specified 51% Female 

Concurrent  1  aMath  423  0.79  Late Spring  
Concurrent  1  aMath  450  0.73  End of Year  
Predictive  1  aMath  257  0.75  Fall to Late Spring  
Predictive  1  aMath  1262  0.70  Winter to Early Spring  
Predictive  1  aMath  450  0.73  Late Spring to End of Year  
Concurrent  K  Teacher Rankings*  226  0.610.90  0.82  Winter 
Concurrent  1  Teacher Rankings*  190  0.670.92  0.78  Winter 
*Note. In the winter during the 20132014 data collection, ten first grade, and eleven Kindergarten teachers voluntarily provided a ranked list of the students in their class. Teachers were encouraged to rank their students based on their perception of their students’ current ability in mathematics. The classrooms were across 4 schools within 2 school districts in the upper Midwest and included 190 First grade and 226 Kindergarten students. A Spearman Rho correlation was calculated within each class between teacher rankings and student performance on the winter earlyMath assessments. Correlations for First grade ranged between r = .61 and .90 and correlations in Kindergarten ranged from r= .67 to .92. The median Spearman Rho correlation was .78 for first grade and .82 (sd = .07) for Kindergarten. In line with previous research on educational assessment (Hoge, & Coladarci, 1989; Gresham, MacMillan, & Bocian, 1997) student scores on the earlyMath assessments were moderately to highly correlated with teacher perceptions of student achievement.
Construct and content validity depend substantially on the development and selection of stimuli. See below for a description of the content.
Decomposing (kindergarten) test construction. There are 8 items representing the decomposing skill, and they are separated into two categories: decomposing from 5, and decomposing from 10. There are four items in each category, and all of the items are presented in either a 5 or 10frame format. For all items, the 5 and ten frames are filled with images of foods (e.g., cookies, apples). During the example item, students are presented with the target number (the “whole”) verbally and visually, as represented by five images in a row. The examiner says, “There are 5 cookies, but I ate 4. How many are left?” The test items are presented in a similar format. Students are presented with dots organized in a 5 or 10frame pattern and the examiner asks, “I ate X, how many are left?” The Decomposing assessment is designed to measure if students automatically know how to decompose numbers to 5 and 10 and are therefore not allowed to use any counting strategies.
Decomposing (first grade) test construction. There are 24 items on this test and 8 target numbers are used, each with 3 opportunities for responses. Target numbers include numbers between 5 and 20. The target number, the “whole,” is located at the top of the student stimulus page, and each of the three items are aligned vertically beneath the target number in a 2 x 1 matrix. The left side of the matrix contains the “part” represented by a numeral or set of dots. If a set of dots was used to represent the “part,” arrays did not contain more than 5 dots. The right side of each matrix is blank, and the student is required to determine what number is the missing “part.” Each of the three given “parts” on the student stimulus sheets were decided based on the following guidelines:
 The “part” is approximately 33% or less of the target number.
 The “part” is between 33 – 66% of the target number.
 The “part” is approximately 66% or greater than the target number.
For example, if the target number is 12, the three given “parts” used as student prompts could be 2, 6, and 9.
Composing test construction. There are 8 items representing the composing skill, and they are separated into two categories: composing to 5, and composing to 10. There are four items in each category, and all of the items are presented in either a 5 or 10frame format. However, boxes were not included to frame the dots to ensure that students are about to compose numbers rather than subitize the blank boxes. During the example item, students are presented with the target number (the “part”) verbally and visually, as represented by dots in a row. The examiner says, “How many more marbles do I need to make five?” The test items are presented in a similar format. Students are presented with dots organized in a 5 or 10frame pattern and the examiner asks, “How many more to make 5” or “How many more to make 10?” The Composing assessment is designed to measure if students automatically know how to compose numbers to 5 and 10 and are therefore not allowed to use any counting strategies.
Story Problems Test Construction. The test consists of six items of two different types: Visual and Verbal. Three are addition and three are subtraction stories. The first two items are visual and the following four are verbal.
Visual items. For the two visual items, students are presented with four possible number expressions and instructed to select the option that corresponds to the visual story problem. In each item, the visual appears at the top of the page and the options are centered and aligned vertically beneath the picture. Only one option correctly corresponds with the visual story problem, and the other three choices are plausible distractors. Criteria for plausible distractors are as follows:
 Number expressions that contain the correct numerals, but have an incorrect symbol (e.g., 2 +4 when the correct response is 2 – 4)
 Number expressions that contain numerals only +/ 1 away from the correct numeral (e.g., 2 + 2 when the correct response is 2 + 3). This allows the examiner to determine if the student is correctly counting the objects in the picture.
 Number expressions that reflect incorrect thinking about the solution (e.g., 5 + 2 when the correct response is 7 – 2, the former response shows that the student may not have realized subtraction was the necessary operation and instead added the answer to the “part” being subtracted).
Verbal items. The following four items are read to the student by the examiner. Students are prompted to solve the problem rather than point to the expression that represents it. For example: “There were 7 cookies on the table. A girl ate 2. How many cookies are on the table now?” The numbers used in all six items were chosen strategically to assess important first grade skills identified by the standards and content experts (teachers, interventionists, math specialists, researchers).
Numeral Identification Kindergarten Test Construction. All numerals between 0 and 31 were used. The test consists of 75 items arranged in 15 rows with 5 numerals per row. The first seven rows of the student sheet are stratified from easier to more difficult items in the following manner: rows one and two contain only single digits 09, rows threesix, and the first two spaces of row seven contain all double digit numerals 1031. Each of the rows mentioned above contain each number within the specified range only once. The remainder of row 7 and rows 815 contain all numbers 031 with some repetition across rows. The order of numerals within each row was determined using a random number generator, but manual changes were made if the random number generator created consecutive sequences of counting or patterns that would be recognized by the student (e.g., 4, 5, 6 or 5, 10, 15). This is a 1 minute timed assessment that also has an inventory option. An inventory option allows the teacher to find out which numerals between 0 and 31 the student has mastered, even if the student does not complete the task in 1 minute.
Numeral Identification First Grade Test Construction. All numerals between 0 and 120 were used. The test consists of 96 items arranged in 16 rows, with 6 numerals per row. The first seven rows of the student sheet are stratified from easier to more difficult items in the following manner: row one contains only single digits, row two contains numerals 1119, row three contains 0, and multiples of 10, row four contains numerals 2129, row five contains numerals 3149, row six contains numerals 5199, and row seven contains numerals 101119. Each of the rows mentioned above contain each number within the specified range only once. Rows 816 contain all numerals 0120 with some repetition across rows. The order of numerals within each row was determined using a random number generator, but manual changes were made if the random number generator created consecutive sequences of counting or patterns that would be recognized by a student (e.g., 4, 5, 6 or 5, 10, 15). This is a 1 minute timed assessment that also has an inventory option. An inventory option allows the teacher to find out which numerals between 0 and 120 the student has mastered, even if the student does not complete the task in 1 minute.
Number Sequence Kindergarten Test Construction. The measure was constructed with 13 items separated by the type of question asked. Types of items include: Count Sequence, Number After, Number Before, and Number Between. Numbers included in the assessment were chosen strategically with the consultation of content experts (teachers, interventionists, math specialists, researchers) to measure important skills.
Count Sequence. The Count Sequence test has two counting forward items and two counting backward items. In each item, the examiner starts the count sequence by saying a consecutive sequence of three numbers (e.g., 2, 3, 4). Item 1 requires the student to count forward to 10 starting from 1, and item 2 requires the student to count forward to 31 starting from a number other than 1, and points are awarded when the student reaches 15, 20, and 31 without error. Item 3 requires the student to count back three numbers from a single digit number between, and Item 4 requires the student to count back five numbers from a number between 8 and 31.
Number After. The Number After test has three items, each one is more difficult than the previous item. The three categories of items are as follows:
 The first prompt is “What number comes after x?”
 The second prompt is “What is one more than x?”
 The third prompt is “What is two more than x?”
Number Before. The Number Before test also has three items, each one more difficult than the previous item. The three categories of items are as follows:
 The first prompt is “What number comes before x?”
 The second prompt is “What is one less than x?”
 The third prompt is “What is two less than x?”
Number Between. The Number Between test has one item that uses the prompt “What number is between x and y?”
Number Sequence First Grade Test Construction. The measure was constructed with 14 items separated by the type of question asked. Consistent with the kindergarten measure, types of items include: Count Sequence, Number After, Number Before, and Number Between. Numbers included in the assessment were chosen strategically with the consultation of content experts (teachers, interventionists, math specialists, researchers) to measure important skills and standards for students in first grade.
Count Sequence. The Count Sequence test has three counting forward items and two counting backward items. In each item, the examiner starts the count sequence by saying a consecutive sequence of three numbers (e.g., 2, 3, 4). Item 1 requires the student to count forward to 20 starting from 14, and item 2 requires the student to count forward to 85 starting from 79. Item 3 requires the student to count forward to 120 from 110. Item 4 requires the student to count back from 55, and Item 5 requires the student to count back to 98 from 102. These items were strategically chosen to be able to assess a number of counting skills such as crossing a decade or knowledge of numbers in the hundreds.
Number After. The Number After test has four items, each one is more difficult than the previous item. The three categories of items are as follows:
 The first prompt is “What number comes after x?”
 The second prompt is “What is one more than x?”
 The third prompt is “What is two more than x?”
 The fourth prompt is “What is ten more than x?”
Number Before. The Number Before test also has four items, each one more difficult than the previous item. The three categories of items are as follows:
 The first prompt is “What number comes before x?”
 The second prompt is “What is one less than x?”
 The third prompt is “What is two less than x?”
 The fourth prompt is “What is ten less than x?”
Number Between. The Number Between test has one item that uses the prompt “What number is between x and y?”
Match Quantity Test Construction. There are 20 items, and this is a 1 minute timed assessment. Each item is organized with an array of blue dots (3/4 in. in diameter) on the left side of the student stimulus page, and a 2 x 2 matrix of numerals on the right side of each page. Dot arrays between 1 and 10 were used for this assessment. Arrays containing 1 to 6 dots have both scattered and ordered configurations, while arrays containing 7 – 10 dots are presented in an ordered configuration.
The matrix containing four numerals always contains the target number and three plausible distractors. Plausible distractors are defined as meeting one of the following criteria:
 One or two numerals greater than the target number
 One or two numerals less than the target number
 A numerals that appeared in the ordered configuration (i.e., in a 3 x 3 ordered configuration of dots, the target number is 9, and a plausible distractor would be 3 or 6 because of the way the dots are presented)
The target number was carefully placed to ensure that it did not appear in the same section of the matrix for each item.
Disaggregated Reliability, Validity, and Classification Data for Diverse Populations
Disaggregated Reliability
Type of Reliability 
Age or Grade 
n (range) 
Coefficient Range 
Coefficient Median 
Information (including normative data)/Subjects 
Alpha 
K 
14 

.95 
American Indian or Alaskan Native 
Alpha 
K 
23 

.97 
Asian or Pacific Islander 
Alpha 
K 
22 

.95 
Hispanic 
Alpha 
K 
46 

.96 
Black 
Alpha 
K 
506 

.95 
White 
Delayed TestRetest (~13 week delay) 
K 
45 

.98 
Asian or Pacific Islander 
Delayed TestRetest 
K 
5 

.86 
Hispanic 
Delayed TestRetest 
K 
89 
.86.90 
.88 
Black 
Delayed TestRetest 
K 
110 
.79.82 
.81 
White 
Alpha 
1 
10 

.80 
American Indian or Alaskan Native 
Alpha 
1 
16 

.92 
Asian or Pacific Islander 
Alpha 
1 
14 

.87 
Hispanic 
Alpha 
1 
29 

.90 
Black 
Alpha 
1 
221 

.85 
White 
Disaggregated Validity
Type of Validity 
Age or Grade 
Test or Criterion 
n (range) 
Coefficient Range 
Coefficient Median 
Information (including normative data)/Subjects 
Concurrent 
KG 
aMath 
1314 
.76.88 
.82 
American Indian Winter & Spring 
Concurrent 
KG 
aMath 
2129 
.69.79 
.74 
Asian or Pacific Islander Winter & Spring 
Concurrent 
KG 
aMath 
123 

.72 
Hispanic Winter 
Concurrent 
KG 
aMath 
55 

.75 
Black Winter 
Concurrent 
KG 
aMath 
742 

.65 
White Winter 
Predictive 
KG 
aMath 
916 
.79.85 
.82 
American Indian Fall to Winter, Fall to Spring, Winter to Spring 
Predictive 
KG 
aMath 
1722 
.74.82 
.78 
Asian or Pacific Islander Fall to Winter, Fall to Spring, Winter to Spring 
Predictive 
KG 
aMath 
17212 
.57.79 
.68 
Hispanic Winter to Spring Early Spring to Late Spring 
Predictive 
KG 
aMath 
49 

.64 
Black Fall to Winter 
Predictive 
KG 
aMath 
778 

.65 
White Early Spring to Late Spring 
Concurrent 
1 
aMath 
14 

.71 
Other (Am Ind, As, Bl, Hisp)* Spring 
Concurrent 
1 
aMath 
263 

.73 
White Spring 
Predictive 
1 
aMath 
10 
.65.91 
.78 
Other (Am Ind, As, Bl, Hisp)* Fall to Spring, Winter to Spring 
Predictive 
1 
aMath 
197260 
.66.74 
.70 
White Fall to Spring, Winter to Spring 
*Note. Samples sizes for American Indian, Asian, Black, and Hispanic populations were too small to compute validity correlations separately for first grade (N = 0 to 4). Many of the schools in the first grade sample did not indicate student ethnicity. These students were identified as “other” or “not specified” and therefore not used in these disaggregate analyses.