Strategic Math Series: Partial Products
Study: Flores & Milton (2020)
Summary
This program contains the materials needed to teach the partial products algorithm for multiplication using the ConcreteRepresentationalAbstract (CRA) method of instruction with an emphasis on the mathematical practices infused throughout the Numbers and Operations standards in most states. State standards call for computation using strategies based on place value. One such strategy is the partial products algorithm. The materials allow for computation instruction within the context of meaningful problem situations. As students master and demonstrate understanding of multiplication, the materials assist them in understanding its relation to other operations. It is intended for use with elementary students with disabilities or who struggle in mathematics. Students who benefitted from its use demonstrated the following deficits: lacked a sense of numbers, did not understand that multidigit numbers are not just separate numerals, but each one has a different value (47 is 4 tens and 2 ones rather than a 4 and 2). Students had attempted to memorize steps to the algorithm without a sense of numbers engage in various type of error patterns. Students who participated demonstrated understanding of addition with regrouping, singledigit multiplication, and showed fluency in multiplication facts 15. The purpose of this program is to build students’ sense of numbers and understanding of the multiplication operation. In addition, the program is about understanding the operation in the context of reallife situations. Therefore, each lesson presents computation problems with words that build into word problems. As lessons progress, students differentiate between addition, subtraction, and multiplication within word problems. This allows students to engage in mathematical practices.
 Target Grades:
 4, 5, 6, 7, 8
 Target Populations:

 Students with learning disabilities
 Any student at risk for academic failure
 Area(s) of Focus:

 Concepts and/or word problems
 Whole number arithmetic
 Comprehensive: Includes computation/procedures, problem solving, and mathematical concepts
 Where to Obtain:
 Kansas Center for Research on Learning
 KUCRL 1122 West Campus Road • Rm. 732 , Lawrence, KS 66045
 (785)8644780
 https://deptsec.ku.edu/~kucrl/catalogsearch/result/?q=math
 Initial Cost:
 $66.00 per manual (paper copy or download)
 Replacement Cost:
 $66.00 per manual (paper copy or download) per N/A

 Staff Qualified to Administer Include:

 Special Education Teacher
 General Education Teacher
 Math Specialist
 Interventionist
 Paraprofessional
 Other:
 Training Requirements:
 Training not required

The manual provides pictorial directions for each problem with specific examples of teacher behavior and think' aloud examples. It is assumed that the interventionist understands elementarylevel mathematics concepts related to numbers, place value, and multiplication. Teacher certification is not a prerequisite.
 Access to Technical Support:
 Professional development support is available through KUCRA. There are traditional professional development sessions, coaching options, and videos available for teachers to view lesson demonstrations
 Recommended Administration Formats Include:

 Individual students
 Small group of students
 Minimum Number of Minutes Per Session:
 25
 Minimum Number of Sessions Per Week:
 2
 Minimum Number of Weeks:
 4
 Detailed Implementation Manual or Instructions Available:
 Yes
 Is Technology Required?
 No technology is required.
Program Information
Descriptive Information
Please provide a description of program, including intended use:
This program contains the materials needed to teach the partial products algorithm for multiplication using the ConcreteRepresentationalAbstract (CRA) method of instruction with an emphasis on the mathematical practices infused throughout the Numbers and Operations standards in most states. State standards call for computation using strategies based on place value. One such strategy is the partial products algorithm. The materials allow for computation instruction within the context of meaningful problem situations. As students master and demonstrate understanding of multiplication, the materials assist them in understanding its relation to other operations. It is intended for use with elementary students with disabilities or who struggle in mathematics. Students who benefitted from its use demonstrated the following deficits: lacked a sense of numbers, did not understand that multidigit numbers are not just separate numerals, but each one has a different value (47 is 4 tens and 2 ones rather than a 4 and 2). Students had attempted to memorize steps to the algorithm without a sense of numbers engage in various type of error patterns. Students who participated demonstrated understanding of addition with regrouping, singledigit multiplication, and showed fluency in multiplication facts 15. The purpose of this program is to build students’ sense of numbers and understanding of the multiplication operation. In addition, the program is about understanding the operation in the context of reallife situations. Therefore, each lesson presents computation problems with words that build into word problems. As lessons progress, students differentiate between addition, subtraction, and multiplication within word problems. This allows students to engage in mathematical practices.
The program is intended for use in the following age(s) and/or grade(s).
Age 35
Kindergarten
First grade
Second grade
Third grade
Fourth grade
Fifth grade
Sixth grade
Seventh grade
Eighth grade
Ninth grade
Tenth grade
Eleventh grade
Twelth grade
The program is intended for use with the following groups.
Students with learning disabilities
Students with intellectual disabilities
Students with emotional or behavioral disabilities
English language learners
Any student at risk for academic failure
Any student at risk for emotional and/or behavioral difficulties
Other
If other, please describe:
Students with Other Health Impairments (ADHD diagnosis) as well as students receiving instruction within MTSS participated in field tests
ACADEMIC INTERVENTION: Please indicate the academic area of focus.
Early Literacy
Alphabet knowledge
Phonological awareness
Phonological awarenessEarly writing
Early decoding abilities
Other
If other, please describe:
Language
Grammar
Syntax
Listening comprehension
Other
If other, please describe:
Reading
Phonics/word study
Comprehension
Fluency
Vocabulary
Spelling
Other
If other, please describe:
Mathematics
Concepts and/or word problems
Whole number arithmetic
Comprehensive: Includes computation/procedures, problem solving, and mathematical concepts
Algebra
Fractions, decimals (rational number)
Geometry and measurement
Other
If other, please describe:
Writing
Spelling
Sentence construction
Planning and revising
Other
If other, please describe:
BEHAVIORAL INTERVENTION: Please indicate the behavior area of focus.
Externalizing Behavior
Verbal Threats
Property Destruction
Noncompliance
High Levels of Disengagement
Disruptive Behavior
Social Behavior (e.g., Peer interactions, Adult interactions)
Other
If other, please describe:
Internalizing Behavior
Anxiety
Social Difficulties (e.g., withdrawal)
School Phobia
Other
If other, please describe:
Acquisition and cost information
Where to obtain:
 Address
 KUCRL 1122 West Campus Road • Rm. 732 , Lawrence, KS 66045
 Phone Number
 (785)8644780
 Website
 https://deptsec.ku.edu/~kucrl/catalogsearch/result/?q=math
Initial cost for implementing program:
 Cost
 $66.00
 Unit of cost
 manual (paper copy or download)
Replacement cost per unit for subsequent use:
 Cost
 $66.00
 Unit of cost
 manual (paper copy or download)
 Duration of license
 N/A
Additional cost information:
Describe basic pricing plan and structure of the program. Also, provide information on what is included in the published program, as well as what is not included but required for implementation (e.g., computer and/or internet access)
Program Specifications
Setting for which the program is designed.
Small group of students
BI ONLY: A classroom of students
If groupdelivered, how many students compose a small group?
28Program administration time
 Minimum number of minutes per session
 25
 Minimum number of sessions per week
 2
 Minimum number of weeks
 4
 If intervention program is intended to occur over less frequently than 60 minutes a week for approximately 8 weeks, justify the level of intensity:
 Lessons last 3045 minutes. The first four lessons may take 45 minutes. There are 18 lessons. In field testing, sessions occurred at least 2 days per week. If only 60 minutes were devoted to the program per week, it would take 9 weeks. Lessons must be mastered prior to moving to the next, so these figures assume that students show mastery. Students within field tests did not repeat lessons.
Does the program include highly specified teacher manuals or step by step instructions for implementation? Yes
BEHAVIORAL INTERVENTION: Is the program affiliated with a broad school or classwide management program?
If yes, please identify and describe the broader school or classwide management program: 
Does the program require technology?  No

If yes, what technology is required to implement your program? 
Computer or tablet
Internet connection
Other technology (please specify)
If your program requires additional technology not listed above, please describe the required technology and the extent to which it is combined with teacher smallgroup instruction/intervention:
Training
 How many people are needed to implement the program ?
 1
Is training for the instructor or interventionist required? No
 If yes, is the necessary training free or atcost?
Describe the time required for instructor or interventionist training: Training not required
Describe the format and content of the instructor or interventionist training:
What types or professionals are qualified to administer your program?
General Education Teacher
Reading Specialist
Math Specialist
EL Specialist
Interventionist
Student Support Services Personnel (e.g., counselor, social worker, school psychologist, etc.)
Applied Behavior Analysis (ABA) Therapist or Board Certified Behavior Analyst (BCBA)
Paraprofessional
Other
If other, please describe:
 Does the program assume that the instructor or interventionist has expertise in a given area?

Yes
If yes, please describe:
The manual provides pictorial directions for each problem with specific examples of teacher behavior and think' aloud examples. It is assumed that the interventionist understands elementarylevel mathematics concepts related to numbers, place value, and multiplication. Beyond foundational understanding of numbers and operations, anyone who can provide instruction to students is qualified.
Are training manuals and materials available? Yes

Describe how the training manuals or materials were fieldtested with the target population of instructors or interventionist and students:  The manual provides pictorial directions for each problem with specific examples of teacher behavior and think' aloud examples. It is assumed that the interventionist understands elementarylevel mathematics concepts related to numbers, place value, and multiplication. Teacher certification is not a prerequisite.
Do you provide fidelity of implementation guidance such as a checklist for implementation in your manual? Yes

Can practitioners obtain ongoing professional and technical support? 
Yes
If yes, please specify where/how practitioners can obtain support:
Professional development support is available through KUCRA. There are traditional professional development sessions, coaching options, and videos available for teachers to view lesson demonstrations
Summary of Evidence Base
 Please identify, to the best of your knowledge, all the research studies that have been conducted to date supporting the efficacy of your program, including studies currently or previously submitted to NCII for review. Please provide citations only (in APA format); do not include any descriptive information on these studies. NCII staff will also conduct a search to confirm that the list you provide is accurate.

Flores, M. M., & Milton, J. H. (2020). Teaching the partial products algorithm using the concreterepresentationalabstract sequence. Exceptionality, 28(2), 142160.
Flores, M. M., Moore, A.J., & Meyer, J. M. (2020) Teaching the partial products algorithm with the concrete representational abstract sequence and the strategic instruction model. Psychology in the Schools, 57(6), 946958.
Flores, M. M., Kaffar, B. J., & Hinton, V. M. (2019). A comparison of CRASIM and direct instruction to teach multiplication with regrouping. International Journal of Research in Learning Disabilities, 4, 2740. Retrieved from http://www.iarld.com/home/thejournalthalamus
Flores, M. M., Schweck, K. B., & Hinton, V. M. (2014). Teaching multiplication with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 29(4), 171183.
Flores, M. M., Hinton, V. M., & Strozier, S. D. (2014). Teaching subtraction and multiplication with regrouping using the concreterepresentationalabstract sequence and strategic instruction model. Learning Disabilities Research and Practice, 29, 7588.
Flores, M. M., & Franklin, T. M. (2014). Teaching multiplication with regrouping using the concreterepresentationalabstract sequence and the strategic instruction model. Journal of American Special Education Professionals,6, 133148.
Study Information
Study Citations
Flores, M. M. & Milton, J. H. (2020). Teaching the partial products algorithm using the concreterepresentationalabstract sequence. . Exceptionality, 28(2) 142160.
Participants
 Describe how students were selected to participate in the study:
 The criteria for participation were as follows: a) mastery of multiplication facts with digits 0–5, b) deficit in multiplication with regrouping as defined as writing less than 20 correct digits in one minute and less than 50% accuracy in obtaining correct products, c) eligibility for special education services with individualized education program goals related to multiplication of multidigit numbers, and d) parent permission for participation.

Describe how students were identified as being at risk for academic failure (AI) or as having emotional/behavioral difficulties (BI):  Two students qualified for special education services under the category of other health impairment because they had a medical condition of Attention Deficit Hyperactivity Disorder that affected attention and learning. The third student qualified for special education services under the category of specific learning disability (SLD). The eligibility criteria for SLD was a 16point or greater difference between predicted and actual achievement. Predicted achievement was obtained using the student’s cognitive ability score and a regression table.

ACADEMIC INTERVENTION: What percentage of participants were at risk, as measured by one or more of the following criteria: below the 30th percentile on local or national norm, or
 identified disability related to the focus of the intervention?
 100.0%

BEHAVIORAL INTERVENTION: What percentage of participants were at risk, as measured by one or more of the following criteria: emotional disability label,
 placed in an alternative school/classroom,
 nonresponsive to Tiers 1 and 2, or
 designation of severe problem behaviors on a validated scale or through observation?
 %
Provide a description of the demographic and other relevant characteristics of the case used in your study (e.g., student(s), classroom(s)).
Case (Name or number)  Age/Grade  Gender  Race / Ethnicity  Socioeconomic Status  Disability Status  ELL status  Other Relevant Descriptive Characteristics 

test  test  test  test  test  test  test  test 
Design
 Please describe the study design:
 The researchers implemented a multiple probe across students design (Horner & Baer, 1978). Each of the students began in baseline. The first student moved from baseline to intervention when baseline data were stable as defined as the last three data points varying no more than 20% from the mean of baseline. Once the first student demonstrated at least 60% accuracy across three probes, the second student moved from baseline to intervention. The third student moved from baseline to intervention when the second student showed an improvement of three probes at or above 60% accuracy. The researchers defined mastery as at least three consecutive probes with at least 30 digits written correctly and all completed problems (100%) had correct products. The researchers collected fluency and accuracy data for each probe because it with possible to write many correct digits across the partial products but fail to arrive at the correct product. Therefore, the researchers intended to ensure that students demonstrated fluency in completing problems correctly. After students completed the intervention, they completed a maintenance probe each week for up to eight weeks. One year after the intervention, the researchers administered another maintenance probe to participants.
Clarify and provide a detailed description of the treatment in the submitted program/intervention: Instructional Procedures Lesson structure. The teacher used explicit instruction during each lesson. This included an advance organizer in which the teacher previewed the lesson and provided expectations for behavior. Next, the teacher modeled by physically showing the problemsolving process and thinking aloud. She involved the students in the process by asking them to repeat information, count with her aloud, or provide answers to known mathematics facts. The third step was guided practice. The teacher and students solved problems in a back and forth process with the teacher completing one part of the problem and the student completing the next part. The teacher provided prompts when needed. After guided practice, the teacher conducted independent practice by giving students problems to complete without her assistance. The last step was a post organizer in which the teacher highlighted the important parts of the lesson and previewed the next lesson. Concrete instruction. The first four lessons were concrete and included the use of base ten blocks, a place value mat, and learning sheet. The teacher read the word problem and talked about what was happening. For example, multiple shelves each had the same number of books and they were finding how many in all by combining groups with the same amount in each. Completed an item about groups (e.g., ___ of ___) by using the problem parts to complete the phrase (e.g., 24 groups 23). Then, the teacher talked about translating that phrase into a multiplication equation and wrote the equation. The teacher implemented the following steps to compute the problem, 24 x 23. She talked about the composition of each number: twentythree included twenty and three and twentyfour had twenty and four. She told the students that she would multiply each of the parts to make partial products. She explained that the word, partial, meant part. She explained that she would make equations with the digit in the ones place of the multiplier. She talked about the commutative property and that she could make four groups of three or three groups of four. She made equal groups on the multiplication mat in the ones place. The answer was 12, so the teacher showed regrouping by exchanging 10 ones for a tens block and placing them on the mat in the appropriate place value column. She wrote 12 in the problem and talked about where to write the numbers, emphasizing that there was one ten in the tens place and two ones in the ones place. On the mat, the next set of rows was shaded gray. She made the equation to find the second partial product with the multipliers 20 and three. She made three groups of two tens blocks (3 x 20). The teacher talked about the product shown with blocks on the mat and wrote 60 in the problem. She referenced the blocks on the mat and talked about how to write 60 with six in the tens place and zero in the ones place. The teacher continued making equations with blocks and writing the other partial products in the problem (80 and 400). The last step was adding all the partial products to find the total product. Beginning with blocks in the ones column and moving to the tens and hundreds, the teacher moved all the blocks in each column to the bottom of the mat. She wrote the sums from each column in the written problem. The teacher checked the answer for reasonableness. She estimated each of the multipliers by rounding to the nearest ten or a familiar number (e.g., 20 x 25), and she multiplied the estimates. She talked about making an easier problem that she could complete mentally to check her work. The estimated answer was 500 and that was close to the actual answer; it was reasonable. Students could not move to the next set of lessons unless the student achieved 100% accuracy on the independent practice problems within the concrete lessons. Figure 2 shows a problem represented on the place value mat using base ten blocks. Representational instruction. Lessons five through eight were representational and involved drawing problems on a place value table printed on the learning sheet. Solving problems involved the same steps as concrete instruction. The teacher and students drew ones as short tallies, tens as long vertical lines, and hundreds as squares. To show regrouping, they circled the drawings. Students could not move to the next set of lessons unless they achieved 100% accuracy on representational lessons. Abstract instruction. The teacher taught the students a mnemonic strategy to help them remember the procedural steps of the partial products algorithm. In lesson nine, students memorized the strategy before using it to solve problems. The strategy was as follows: (a) Read the problem; (b) Examine the ones in the multiplier to make equations; (c) Note the partial products for ones; (d) Address the tens in the multiplier to make equations; (e) Mark the partial products for tens; and (f) Examine the columns, add, and check (RENAME). It is important to note that the second step prompted students to examine the ones in the multiplier and think about the two equations that would be made. The third step of the strategy prompted students to execute the operations and write the two partial products generated from those equations in the problem. The fourth and fifth steps prompted similar responses but using the tens place of the multiplier. The processes associated with the RENAME strategy are shown in Figure 3. There was one lesson in which the students applied RENAME to multiplication equations. In order to prepare students for discrimination among operations, the teacher taught students an additional strategy. The strategy was as follows: (a) Find what you are solving for; (b) Ask yourself, “What are the parts of the problem;” (c) Set up the numbers; and (d) Tie down the sign. The students learned to find question within the word problem. The next step was thinking about that was happening in the problem by finding the parts of the problem. The teacher taught the students to notice whether objects came together or separated. If objects came together, they noticed whether groups of the same size combined (multiplication) or groups of different sizes combined (addition). The students recognized subtraction as separation of a group. Then, they used the parts of the problem to identify the numbers that would be used, they confirmed the operation and wrote the equation. The students completed lessons in which they solved word problems that required addition, subtraction, or multiplication. When the problems involved multiplication, they used the two strategies, FAST RENAME. They also completed equations using RENAME.
Clarify what procedures occurred during the control/baseline condition (third, competing conditions are not considered; if you have a third, competing condition [e.g., multielement single subject design with a third comparison condition], in addition to your control condition, identify what the competing condition is [data from this competing condition will not be used]): During baseline, students received no instruction related to multiplication. The teacher administered probes (same as those used in intervention phases). The probes were timed for two minutes. After one minute, the teacher provided no feedback and thanked the student for participating. Each student worked with the teacher oneonone. Students in baseline had no exposure to the intervention because their scheduled instruction occurred at different times of the day.
Please describe how replication of treatment effect was demonstrated (e.g., reversal or withdrawal of intervention, across participants, across settings) Multiple probe across students.

Please indicate whether (and how) the design contains at least three demonstrations of experimental control (e.g., ABAB design, multiple baseline across three or more participants).  There were three demonstrations of of effect at three different points in time. The decision to move from baseline to treatment was responsebased. For the first student, treatment began after stable baseline (stable baseline, last three data points varying no more than 20% from the mean of baseline). When student one reached 60% accuracy cross three probes, student two began intervention if baseline data were stable. The same procedures occurred for student three. The criteria for mastery of at least 3 consecutive probes with 30 correct digits and 100% accuracy in products. Each student, at a different point in time reached this mastery criterion.
If the study is a multiple baseline, is it concurrent or nonconcurrent? Nonconcurrent
Fidelity of Implementation
 How was the program delivered?

Individually
Small Group
Classroom
If small group, answer the following:
 Average group size
 Minimum group size
 Maximum group size
What was the duration of the intervention (If duration differed across participants, settings, or behaviors, describe for each.)?
 Weeks
 6.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 30.00
 Weeks
 6.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 30.00
 Weeks
 6.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 30.00
 What were the background, experience, training, and ongoing support of the instructors or interventionists?
 Special education teacher (the students' teacher). She had a Master’s degree in special education and eight years of teaching experience. The researcher provided the teacher with professional development prior to the intervention. The researcher held a 1.5hour session in which she modeled lessons and the teacher practiced lessons using a printed manual with written and pictorial directions. The teacher demonstrated 100% accuracy in implementation according to a fidelity checklist. After the session, the teacher received the manual as well as videos that showed implementation of lessons at concrete and representational levels as a backup support.
Describe when and how fidelity of treatment information was obtained. The researcher recorded 30% of all lessons across the concrete, representational, and abstract phases according to the recommendations for highquality singlecase research (Kratochwill et al. 2013). Two researchers watched the videos with treatment fidelity checklists. The checklists had teacher behaviors that corresponded to implementing concrete, representational, or abstract instruction. The items had two answers, yes or no. After the researchers watched videos and completed checklists, they compared their lists and calculated interrater agreement with the agreements divided by the total agreements and disagreements.
What were the results on the fidelityoftreatment implementation measure? Treatment fidelity was 97.8%. Interrater agreement was 98.9%.
Was the fidelity measure also used in baseline or comparison conditions? yes
Measures and Results
Measures Broader :
Study measures are classified as targeted, broader, or administrative data according to the following definitions:

Targeted measures
Assess outcomes, such as competencies or skills, that the program was directly targeted to improve. In the academic domain, targeted measures typically are not the very items taught but rather novel items structured similarly to the content addressed in the program. For example, if a program taught wordattack skills, a targeted measure would be decoding of pseudo words. If a program taught comprehension of causeeffect passages, a targeted measure would be answering questions about causeeffect passages structured similarly to those used during intervention, but not including the very passages used for intervention.
 In the behavioral domain, targeted measures evaluate aspects of external or internal behavior the program was directly targeted to improve and are operationally defined.

Broader measures
Assess outcomes that are related to the competencies or skills targeted by the program but not directly taught in the program. In the academic domain, if a program taught wordlevel reading skill, a broader measure would be answering questions about passages the student reads. If a program taught calculation skill, a broader measure would be solving word problems that require the same kinds of calculation skill taught in the program.
 In the behavioral domain, if a program taught a specific skill like ontask behavior in one classroom, a broader measure would be ontask behavior in another setting.
 Administrative data measures apply only to behavioral intervention tools and are measures such as office discipline referrals (ODRs) and graduation rates, which do not have psychometric properties as do other, more traditional targeted or broader measures.
Targeted Measure  Reverse Coded?  Evidence  Relevance 

Targeted Measure 1  Yes  A1  A2 
Broader Measure  Reverse Coded?  Evidence  Relevance 

Broader Measure 1  Yes  A1  A2 
Administrative Data Measure  Reverse Coded?  Relevance 

Admin Measure 1  Yes  A2 
 If you have excluded a variable or data that are reported in the study being submitted, explain the rationale for exclusion:
Results
 Describe the method of analyses you used to determine whether the intervention condition improved relative to baseline phase (e.g., visual inspection, computation of change score, mean difference):
 visual inspection, metric of magnitude of change (TauU)
Please present results in terms of within and between phase patterns. Data on the following data characteristics must be included: level, trend, variability, immediacy of the effect, overlap, and consistency of data patterns across similar conditions. Submitting only means and standard deviations for phases is not sufficient. Data must be included for each outcome measure (targeted, broader, and administrative if applicable) that was described above. Jack. Jack’s baseline for digits correct had a level of 9.14 with a range from six to 11 digits. Jack’s baseline for percent correct had a level of 0%. After the introduction of the intervention, there was an immediate change in both the digits correct (from 10 digits to 24 digits) and percent correct (from 0% to 50%). The level for digits correct was 69 with a range from 24 to 117 digits. There were no overlapping data points; the percentage of nonoverlapping data (PND) was 100%. The level for percent correct 83% with a range from 50% to 100%. There were no overlapping data points; the percentage of nonoverlapping data (PND) was 100%. Jack completed a maintenance probe every week after instruction ended. Jack maintained performance for eight weeks after instruction. The digits correct for maintenance probes ranged from 85 correct digits to 117 correct digits and all of the probes were 100% correct. One year after instruction ended, Jack completed a maintenance probe. He wrote 130 correct digits and completed the problems with 100% accuracy. The researchers conducted TauU. In addition to PND, the researchers calculated TauU. Tauu for Jack was strong for digits correct and percent correct with each at 1.0. Anna. Anna’s baseline for digits correct had a level of 6.33 with a range from four to nine digits. Anna’s baseline for percent correct had a level of 0%. After the introduction of the intervention, there was an immediate change in both the digits correct (from four digits to 22 digits) and percent correct (from 0% to 100%). The level for digits correct was 60.73 with a range from 22 to 88 digits. There were no overlapping data points; the percentage of nonoverlapping data (PND) was 100%. The level for percent correct 96.5% with a range from 50.4% to 100%. There were no overlapping data points; the PND was 100%. Anna completed a maintenance probe every week after instruction ended. Anna maintained performance for six weeks after instruction. The digits correct for maintenance probes ranged from 66 correct digits to 86 correct digits and all the probes were 100% correct. One year after instruction, Anna completed a maintenance probe. She wrote 127 correct digits and completed the probe with 90% accuracy. Tauu for Anna was strong for digits correct and percent correct with each at 1.0. Hugo. Hugo’s baseline for digits correct had a level of 4.2 with a range from two to seven digits. Hugo’s baseline for percent correct had a level of 0%. After beginning intervention, there was an immediate change in digits correct (from five digits to 12 digits). The level for digits correct was 39.7 with a range from 10 to 69. There were no overlapping data points; the PND was 100%. The level for percent correct 68.7% with a range from 0% to 100%. The data path showed an increasing trend. There were three overlapping data points; the PND was 80%. Hugo completed a maintenance probe every week after instruction ended. Hugo maintained performance for three weeks after instruction. The digits correct on the maintenance probe ranged from 55 correct digits to 70 correct digits and all the probes were 100% correct. Hugo did not complete a probe at the oneyear mark because he was not enrolled at the same school. Tauu for Hugo was strong for digits correct with TauU at 1.0. TauU was strong for percent correct at 0.80.
Additional Research
 Is the program reviewed by WWC or EESSA?
 No
 Summary of WWC / EESSA Findings :
 What Works Clearinghouse Review
This program was not reviewed by the What Works Clearinghouse.
Evidence for ESSA
This program was not reviewed by Evidence for ESSA.
 How many additional research studies are potentially eligible for NCII review?
 1
 Citations for Additional Research Studies :
Flores, M. M., & Franklin, T. M. (2014). Teaching multiplication with regrouping using the concreterepresentationalabstract sequence and the strategic instruction model. Journal of American Special Education Professionals, 6, 133148.
Data Collection Practices
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