Strategic Math Series: Standard Algorithm
Study: Flores, Hinton, & Strozier (2014)
Summary
This program contains the materials needed to teach the standard algorithm for multiplication with regrouping using the ConcreteRepresentationalAbstract (CRA) method of instruction with an emphasis on the mathematical practices infused throughout the Numbers and Operations standards in most states. The materials allow for computation instruction within the context of meaningful problem situations. As students master and demonstrate understanding of multiplication with regrouping, the materials assist them in understanding its relation to other operations. The program is intended for elementary or middle level students who struggle. Students with and without disabilities who participated in instruction showed benefit made errors in baseline assessments that showed: poor sense of numbers, lacked understanding 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). Participating students had attempted to memorize steps to the standard algorithm without a sense of numbers and engaged in various type of error patterns. 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. So, each lesson presents computation problems with words. These build into word problems and finally, students differentiate between addition, subtraction, and multiplication problems. This allows students to engage in mathematical practices.
 Target Grades:
 5, 6, 7, 8
 Target Populations:

 Students with learning disabilities
 Any student at risk for academic failure
 Other: Students with disabilities such as Other Health Impairments, Autism Spectrum Disorder participated in research as well as students receiving instruction within MTSS
 Area(s) of Focus:

 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

Teachers will need sets of base ten blocks. Two sets will be plenty to model problems; the reason there is a need for 2 sets is because some problems will require 60 tens blocks. Within research, students had their own blocks. In field testing, students shared blocks for independent practice.
 Staff Qualified to Administer Include:

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

The manual provides pictorial directions for each problem (step by step) with specific examples of teacher behavior and think' aloud examples. There are "teacher tips" included for trouble shooting based on field testing experiences. The manuals were revised and refined over time to be increasingly userfriendly. Teachers used the manual and demonstrated lessons to researchers who evaluated their performance using a fidelity checklist.
 Access to Technical Support:
 Professional development is available through KUCRL. There are traditional information sessions, coaching options, and videos that show lesson demonstrations
 Recommended Administration Formats Include:

 Small group of students
 Minimum Number of Minutes Per Session:
 30
 Minimum Number of Sessions Per Week:
 3
 Minimum Number of Weeks:
 6
 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 standard algorithm for multiplication with regrouping using the ConcreteRepresentationalAbstract (CRA) method of instruction with an emphasis on the mathematical practices infused throughout the Numbers and Operations standards in most states. The materials allow for computation instruction within the context of meaningful problem situations. As students master and demonstrate understanding of multiplication with regrouping, the materials assist them in understanding its relation to other operations. The program is intended for elementary or middle level students who struggle. Students with and without disabilities who participated in instruction showed benefit made errors in baseline assessments that showed: poor sense of numbers, lacked understanding 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). Participating students had attempted to memorize steps to the standard algorithm without a sense of numbers and engaged in various type of error patterns. 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. So, each lesson presents computation problems with words. These build into word problems and finally, students differentiate between addition, subtraction, and multiplication 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 disabilities such as Other Health Impairments, Autism Spectrum Disorder participated in research as well as students receiving instruction within MTSS
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
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)
Teachers will need sets of base ten blocks. Two sets will be plenty to model problems; the reason there is a need for 2 sets is because some problems will require 60 tens blocks. Within research, students had their own blocks. In field testing, students shared blocks for independent practice.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?
210Program administration time
 Minimum number of minutes per session
 30
 Minimum number of sessions per week
 3
 Minimum number of weeks
 6
 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 3 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. Teacher certification is not a prerequisite.
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 (step by step) with specific examples of teacher behavior and think' aloud examples. There are "teacher tips" included for trouble shooting based on field testing experiences. The manuals were revised and refined over time to be increasingly userfriendly. Teachers used the manual and demonstrated lessons to researchers who evaluated their performance using a fidelity checklist.
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 is available through KUCRL. There are traditional information sessions, coaching options, and videos that show 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., 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., & Hinton, V. M. (2019). Improvement in elementary students’ multiplication skills and understanding after learning through the combination of the concreterepresentationalabstract sequence and strategic instruction. Education and Treatment of Children, 42(1), 7396.
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., 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(1) 7588.
Participants
 Describe how students were selected to participate in the study:
 The students met the criteria for participation which were: (a) parent permission, (b) mastery of addition, subtraction, and multiplication facts as defined as writing 30 correct digits per minute on a curriculumbased measure, (c) demonstration of knowledge of place value as demonstrated by correct identification of ones’, tens’, and hundreds’ places, and lack of skill associated with subtraction and multiplication with regrouping as defined by writing less than 10 correct digits on a timed curriculumbased measure.

Describe how students were identified as being at risk for academic failure (AI) or as having emotional/behavioral difficulties (BI):  All were students receiving tier three mathematics intervention within a MTSS model. None of the students had identified disabilities; however, all of the students failed to respond to tier one and tier two mathematics interventions according to benchmark assessments

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:
 multiple probe across behaviors
Clarify and provide a detailed description of the treatment in the submitted program/intervention: Materials The materials consisted of assessment probes and instructional items. There were four sets of assessments probes, one for each behavior: subtraction of threedigit numbers with regrouping in the ones place, b) subtraction of threedigit numbers with regrouping in the ones and tens places, c) multiplication of twodigit and onedigit numbers with regrouping, and d) multiplication of two twodigit numbers with regrouping. Each assessment probe was an eightinch by eleveninch sheet of paper with twentyfour problems printed in twelvepoint font. Measures to ensure content validity for the probes were conducted for each behavior. Pools of the computation questions for each skill area were distributed to three teachers to review. All three teachers had at least a master’s degree from an accredited university and at least four years of experience in teaching mathematics to elementary or middle school students. The reviewers were asked to score each problem for each skill area according to its relevance to the content using a 4point scale to avoid having neutral and ambivalent midpoint. The item relevance continuum used was 1= not relevant, 2= somewhat relevant, 3= quite relevant, and 4= highly relevant. Then for each item, the itemlevel content validity index (ICVI) was computed as the number of experts giving a rating of 3 or 4 (thus dichotomizing the ordinal scale into relevant and not relevant), divided by the total number of experts (Lynn, 1976). The ICVI for all items for subtraction with regrouping of the ones column and for subtraction with regrouping of the ones and tens column was calculated as 1.00 and 1.00 respectively. The ICVI for 48 of the 50 two digit by one digit multiplication with regrouping problems was 1.00 and .66 for the remaining two problems. The ICVI for 63 of the 65 two digit by two digit multiplication with regrouping problems was 1.00 and .66 for two remaining problems. To further examine the items in each probe, the three teachers were also asked to rate the same problems as the following: easy, average or difficult for the typical elementary school student who was learning the computation problem represented by all four behaviors. The difficulty level of the questions required consistency of two of the three raters on each computation problem. Based on the survey, all 56 problems for subtraction with regrouping of the ones column, all 40 problems for subtraction with regrouping of the ones and tens column, and all 50 problems for two digit by one digit multiplication with regrouping were rated as average. The 64 of the 65 problems involving two digit by two digit multiplication with regrouping were rated as average and one problem was rated as easy. Reliability of the probes was also examined for each skill area. The researchers created an assessment for each skill area and each assessment contained 75% or more of the problems from the probes created for the participants. The researchers administered the four assessments to 12 college level students from a major fouryear university. The assessments were untimed and all 12 of the assessments were completed. Results from the internal consistency test revealed Cronbach’s Alpha Coefficient of r = .69 for subtraction with regrouping of the ones column, r = .72 for subtraction with regrouping of the ones and tens column, r = .78 for two digit by one digit multiplication with regrouping and r = .73 for two digit by one digit multiplication with regrouping. There was a Cronbach’s Alpha Coefficient of r = .83 for all test items. The instructional materials varied based on skill and level of instruction. Materials used to teach subtraction with regrouping in the ones place included the following: a) lessons one three through included student learning sheets with subtraction with regrouping problems divided into three sections labeled demonstration, guided practice, and independent practice and foam baseten mathematics manipulative blocks; b) lessons four through six included student learning sheets with subtraction with regrouping problems divided into three sections labeled demonstration, guided practice, and independent practice; c) lesson seven consisted of a learning sheet with the mnemonic RENAME printed with the steps for the RENAME strategy printed in twentypoint font; and d) lessons eight through ten included student learning sheets with subtraction with regrouping problems divided into three sections labeled demonstration, guided practice, and independent practice. Materials used to teach subtraction with regrouping in the ones and tens places, multiplication with onedigit multipliers with regrouping and multiplication with twodigit multipliers with regrouping were similar. These materials included learning sheets and baseten manipulatives for lessons one through three, learning sheets for lessons four through six, a RENAME strategy sheet for lesson seven, and learning sheets for lessons eight through ten. Procedures The first author, a certified special education teacher, served as the instructor. Prior to instruction, probes were administered until the students demonstrated a stable baseline. A stable baseline was defined as the last three data points in the path varying no more than twentypercent from the mean. After stable baselines were established, probes were administered prior to daily instruction in order to measure students’ progress. Probes were administered prior to instruction in order to measure learning that had occurred during the previous day’s instruction. The researcher administered the probes by providing the students with a sheet of problems with instructions to complete as many problems as he/she could until told to stop. The students were given two minutes to complete the problems. Instruction began with subtraction with regrouping in the ones place. Prior to lesson one, the instructor showed each student one of his/her baseline probes, discussed his/her difficulty with computation and obtained a written commitment to participate in learning. The instructor and students agreed to work hard to learn a new way of completing math problems. The instructor continued to involve the students in progress monitoring by showing each student his/her progress on the probes in the form of a graph. The instructor showed the students their graphs individually at the beginning of each instructional session. Each graph included an aim line representing the criterion for each phase so that the students could see their progress toward the goal of thirty correct digits. During lessons one through three, the instructor began the lesson with an advance organizer, providing an overview of the lesson activities. Next, the instructor demonstrated how to solve the problems within the Demonstration section of the learning sheet using baseten manipulatives. This demonstration included physical demonstration as well as thinking aloud. The students participated by answering simple questions in order to maintain engagement, but did not solve problems. The third part of the lesson was guided practice in which the instructor and the students solved problems in the Guided Practice section of the learning sheet together. The instructor and the students alternated turns completing steps of the problem solving process using baseten manipulatives to solve the problems. Next, the students completed the Independent Practice section of the learning sheet using the baseten manipulatives without the instructor’s assistance. Finally, the lesson ended with a post organizer in which the instructor reviewed the lesson activities. During lessons four through six, the instructor began the lesson with an advance organizer, providing an overview of the lesson activities. Next, the instructor demonstrated how to solve the problems within the Demonstration section of the learning sheet by drawing pictures. This demonstration included physical demonstration as well as thinking aloud. The students participated by answering simple questions in order to maintain engagement, but did not solve problems. The third part of the lesson was guided practice in which the instructor and the students solved problems in the Guided Practice section of the learning sheet together. The instructor and the students alternated steps of the problem solving process drawing pictures to solve the problems. Next, the students completed the Independent Practice section of the learning sheet drawing pictures without the instructor’s assistance. Finally, the lesson ended with a post organizer in which the instructor reviewed the lesson activities. During lesson seven, the instructor provided an advance organizer, explaining that the students would learn a memory device to help them remember the steps for solving regrouping problems. The instructor recited the strategy and guided the students in reciting the steps. The students practiced the RENAME steps until they could recite the steps without assistance. After the students had memorized the RENAME mnemonic, lessons eight through ten involved instruction solving problems using numbers only and the RENAME mnemonic device. Lessons eight through ten included an advance organizer, demonstration, guided practice, independent practice, and a post organizer. There were ten lessons that followed this same framework for each of the other behaviors: subtraction with regrouping in the ones and tens places, multiplication with a onedigit multiplier and regrouping, and multiplication with a twodigit multiplier and regrouping.
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]): The baseline condition involved administration of probes with no instruction. The researcher presented student with probe, started a timer for 2 minutes. After 2 minutes, took the probe away and provide not feedback, but thanked the student for working hard.
Please describe how replication of treatment effect was demonstrated (e.g., reversal or withdrawal of intervention, across participants, across settings) replicated across behaviors : subtraction with regrouping in ones, subtraction with regrouping in ones and tens, multiplication with regrouping 1digit multiplier, multiplication with regrouping with 2digit multiplier

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).  A multipleprobe across behaviors design, effects shown across 3 behaviors (Lena & May) and 4 behaviors (Al) at different points in time
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
 3
 Minimum group size
 3
 Maximum group size
 3
What was the duration of the intervention (If duration differed across participants, settings, or behaviors, describe for each.)?
 Weeks
 4.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 30.00
 Weeks
 4.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 30.00
 Weeks
 4.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 30.00
 What were the background, experience, training, and ongoing support of the instructors or interventionists?
 Researchers were interventionists. Both had PhDs, were university faculty, former special education teachers (48 years of teaching experience in K12) who maintained their special education teaching certificates.
Describe when and how fidelity of treatment information was obtained. Procedural integrity data were collected throughout the study. The instructor completed a checklist of lesson activities during instructional lessons. The second author observed lessons live one time per week, for a total of twentyfive percent of lessons and instructional phases.
What were the results on the fidelityoftreatment implementation measure? The observer’s checklists indicated 100% integrity. The two (observer’s and instructor’s) checklists for corresponding dates were compared to compute interobserver agreement which was 100%.
Was the fidelity measure also used in baseline or comparison conditions? both conditions
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 analysis, 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. Baseline for Al, Lena, and May For problems involving regrouping for the ones place, Al wrote an average of 10.6 correct digits with a range from nine to ten correct digits. Lena wrote zero correct digits across all probes and May wrote an average of two correct digits, ranging from zero to three correct digits. For problems involving regrouping for the ones and tens place, Al wrote an average of 3.8 correct digits with a range from three to four correct digits. Lena wrote 9.8 correct digits, ranging from nine to eleven correct digits. May wrote an average of 10.8 correct digits, ranging from five to twelve correct digits. For problems the included multiplication with regrouping with onedigit multipliers, Al wrote zero correct digits across all probes. Lena wrote zero correct digits across all probes. May wrote an average of 5.5 correct digits, ranging from four to six correct digits. For problems that involved multiplication with regrouping with twodigit multipliers, Al wrote zero correct digits across all baseline probes. Neither Lena nor May reached instruction involving multiplication with regrouping with twodigit multipliers. Intervention Performance for Al During intervention with problems that included subtraction with regrouping for the ones place, Al demonstrated an increasing data path with a level of 26.4 correct digits. There was one overlapping data point between phases with the first probe, 11% overlap. Al reached criterion after nine probes. Two weeks and four weeks after instruction in subtraction with regrouping in the ones place, Al wrote thirty correct digits on maintenance probes. During CRA intervention with subtraction problems that required regrouping in the ones and tens places, Al demonstrated an increasing data path with a level of 21.2 correct digits. There were no overlapping data points between intervention and baselines phases. Al reached criterion after eleven probes. Two weeks after instruction in subtraction with regrouping in the ones and tens places, Al wrote thirtyone correct digits, maintaining his performance. Within the multiplication with regrouping with onedigit multiplier phase, Al demonstrated an increasing data path with a level of 41.6 correct digits. There were no overlapping data points between intervention and baseline phases. Al reached criterion after five probes. Two weeks after instruction in multiplication with regrouping with onedigit multipliers, Al wrote sixtyfive correct digits, maintaining his performance. During the last intervention phase, multiplication with regrouping with twodigit multipliers, Al demonstrated an increasing data path with a level of 23.5 correct digits. However, there was one overlapping data point, 17% overlap between baseline and intervention phases. Al reached criterion after six probes. No maintenance data were collected after instruction in multiplication with regrouping with twodigit multipliers because the school year ended. Intervention Performance for Lena During intervention with problems that involved subtraction with regrouping in the ones place, Len demonstrated an increasing data path with a level of 21 correct digits, ranging from eight to fortyone correct digits. There were no overlapping data points between baseline and intervention phases and Lena reached criterion after ten probes. Two weeks and four weeks after instruction in subtraction with regrouping in the ones place, Lena wrote thirtysix and thirtyfive correct digits, maintaining her performance from intervention. Within intervention in subtraction with regrouping in the ones and tens places, Lena demonstrated an increasing data path with a level of 27.8, ranging from fifteen to thirtythree correct digits. There were no overlapping data points between phases and Lena reached criterion after eight probes. Two weeks after instruction in subtraction with regrouping in the ones and tens places, Lena wrote twentyfive correct digits. During intervention in multiplication with regrouping with onedigit multipliers, Lena demonstrated an increasing data path with a level of 27.4 correct digits, ranging from six to fortyfive correct digits. There were no overlapping data points between phases and criterion was reached after seven probes. Maintenance probes for multiplication were not administered because the school year ended. Intervention Performance for May During intervention for problems including regrouping in the ones place, May’s data show an increasing path with some variance. May’s level of performance was 23.8 correct digits with a range from eight to thirtytwo correct digits. There was no overlap between May’s baseline and intervention phases and she met criterion after eleven probes. Two weeks and four weeks after instruction in subtraction with regrouping in the ones place, May wrote twentyfive correct digits one each maintenance probe. During intervention in subtraction with regrouping in the ones and tens places, May demonstrated a steadily increasing trend with a level of 20.9 and a range of seven to thirty correct digits. There was one overlapping data point between baseline and intervention phases, 6% overlap between phases. May met criterion after eighteen probes. Two weeks after instruction, May wrote eighteen correct digits on a probe that included subtraction with regrouping in the ones and tens places. During the last intervention phase, multiplication with regrouping with onedigit multipliers, May demonstrated an increasing data path with some variability. The level of May’s data path was 23.7 with a range from eight to fortytwo correct digits. There was one overlapping data point, 14% overlap between phases. May met criterion after seven probes. Maintenance probes were not administered for multiplication since the school year ended. Effect Size TauU was calculated for each student; this form of analysis combined nonoverlapping data points between phases with trend within the intervention phase while accounting for any trend within baseline (Parker, Vannest, Davis, & Sauber, 2011). For Al, there were no significant trends within baseline phases. In comparing baseline and intervention phases for subtraction with regrouping in the ones place, a strong effect was indicated (TauU=0.97). In comparing AL’s baseline and intervention data for subtraction with regrouping in the ones and tens places, a strong effect was indicated (TauU=1.0). The comparison of Al’s baseline and intervention phases for multiplication with a onedigit multiplier indicated a moderate effect (TauU=1.0). The comparison of Al’s baseline and intervention phases for multiplication with a twodigit multiplier indicated a strong effect (TauU=0.83). Overall, the intervention had a strong effect across all phases (TauU= 0.95). For Lena, there were no significant trends within baseline phases. In comparing baseline and intervention phases for subtraction with regrouping in the ones place, a strong effect was indicated (TauU=1.0). In comparing Lena’s baseline and intervention data for subtraction with regrouping in the ones and tens places, a strong effect was indicated (TauU=1.0). The comparison of Lena’s baseline and intervention phases for multiplication indicated a strong effect (TauU=1.0). Overall, the intervention had a strong effect across all phases (TauU= 1.0). For May, there were no significant trends within baseline phases. In comparing baseline and intervention phases for subtraction with regrouping in the ones place, a strong effect was indicated (TauU=1.0). In comparing May’s baseline and intervention data for subtraction with regrouping in the ones and tens places, a strong effect was indicated (TauU=0.91). The comparison of May’s baseline and intervention phases for multiplication indicated a moderate effect (TauU=0.67). Overall, the intervention had a strong effect across all phases (TauU= 0.86)
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?
 2
 Citations for Additional Research Studies :
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., & 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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