Strategic Math Series: Standard Algorithm
Study: Flores, Hinton, & Schweck (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. & Schweck, K. B. (2014). Teaching multiplication with regrouping to students with learning disabilities. . Learning Disabilities Research & Practice, 29(4) 171183.
Participants
 Describe how students were selected to participate in the study:
 The students were chosen based on the following: a) were eligible for special education services; b) were fluent in basic addition, subtraction, and multiplication as defined as writing at least 45 correct digits per minute; and c) demonstrated mastery of multiplication with regrouping with problems that included a onedigit multiplier; and d) inability to compute multiplication with regrouping problems that included a twodigit multiplier. All of the students met these criteria and demonstrated similar deficits in their ability to complete problems with twodigit multipliers. The students partially completed these problems by attending to the digit in the ones place of the multiplier; error patterns varied.

Describe how students were identified as being at risk for academic failure (AI) or as having emotional/behavioral difficulties (BI): 
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 students
Clarify and provide a detailed description of the treatment in the submitted program/intervention: Instructional materials consisted of the following: a) an instructional manual created by the first author that provided instructional procedures and suggested scripts for each lesson, b) student learning sheets for each lesson, c) place value mats, and d) base ten blocks. The student learning sheets were divided into three sections with problems that were used for teacher demonstration, guided practice, and independent practice. The learning sheet for lesson seven differed from the others; it had the RENAME strategy printed in the middle of the page. The place value mats were laminated pages used to organize base ten blocks or drawings when solving problems at the concrete and representational levels. The place value mat for concrete level instruction was a 36x36 inch sheet which provided the student with space to organize base ten blocks. The place value mat for representational level instruction was a 16x16 inch sheet which provided the student with space to organize drawings while solving problems. Both types of place value mats were tables that had columns labeled for the ones, tens, hundreds, and thousands places. The columns were divided into three rows, each shaded to visually differentiate multiplication by the multiplier digit in the ones place, tens place, and total when both rows were added. The cells were further divided into nine smaller cells for grouping. Procedures Prior to instruction, probes were administered to each student. The teacher placed the probe in front of the student and told him/her to complete as many problems as he/she could until told to stop. The teacher told the student to begin, started a timer, and asked the student to stop after two minutes. Baseline procedures involved the administration of probes only. Instruction began when the first student demonstrated a stable baseline, as defined as at least five data points with the last three data points varying no more than 20% from the mean of the baseline data path. The other students remained in baseline until the first student wrote at least 25 correct digits on a probe. This criteria for phase change was chosen because it would show a clear increase over baseline and the time for the study was limited since it was implemented when other activities (e.g., annual testing, field trips, festivals) and breaks may have interfered with instruction across multiple students if the criteria was thirty correct digits (mastery). Across all phases of instruction, lessons were implemented according to the explicit instruction model with an advance organizer, teacher demonstration, guided practice, independent practice, and a post organizer. The first three lessons involved instruction at the concrete level using base ten blocks. The problem solving steps were as follows. The teacher reviewed to reverse rule or commutative property; employing this property allowed for more efficient computation and manipulation of objects (e.g., four groups of twenty rather than twenty groups of four). The problem was read aloud and the multiplicand (top number) was represented on the place value mat. Beginning in the ones column, the problem was represented, making the appropriate number of groups. The answer was evaluated and if there were ten or more ones, regrouping occurred by exchanging ten ones for one tens block which was placed in the tens column. Regrouping was also noted on the written problem. Next, the tens place of the multiplicand was multiplied by the ones place of the multiplier and this problem was represented on the place value mat. The answer was evaluated and if there were ten or more tens, regrouping occurred by exchanging ten tens for one hundreds block which was placed in the hundreds column. The numbers were noted in the written problem. Before multiplying by the tens place of the multiplier, the teacher/student crossed out the number in the ones place of the multiplier and wrote a zero in the ones place underneath the first line of answers within the problem. Next, number in the ones place of the multiplicand was multiplied by the number in the tens place of the multiplier. The commutative property was employed for ease of problem solving. The problem was represented using base ten blocks. The answer was evaluated for regrouping and if there were ten or more, ten tens were exchanged for a hundreds block which was placed in the hundreds place on the mat. Regrouping was noted on the written problem. The numbers in the tens places of the multiplicand and the multiplier were multiplied (e.g., thirty groups of twenty). This was an unwieldy problem using manipulatives, so it was solved in this way one time for demonstration and then the following short cut was taught. It was demonstrated that thirty groups of twenty was the same as two groups of three hundred. The teacher demonstrated this concept, guided the students in demonstrating the concept, and the students demonstrated this independently. After this instruction, the students were convinced that using hundreds blocks to solve this problem was easier than using many groups of tens blocks. Once the problem was represented and solved, the answer was evaluated for regrouping. If there were ten or more hundreds, they were exchanged for a thousands block. The numbers were noted on the written problem. Finally, the answers obtained by multiplying by both numbers in the multiplier were added. Each row within the columns of the place value mat was added and the numbers within the written problem were added. The final step was to compare the manipulatives within each column of the mat with the answer in the written problem. The procedures for instruction at the representational level (lessons four through six) involved the use of drawings rather than base ten blocks. Ones were drawn by writing small tallies on a horizontal line. Tens were drawn using long vertical lines. Hundreds were drawn as squares and thousands were drawn as cubes. With the exception of the ones, the drawings were similar in form to the base ten blocks. The seventh lesson involved teaching the student a strategy for solving regrouping problems. The strategy was: a) Read the problem; b) Examine the ones; c) Note the ones; d) Address the tens; e) Mark the tens; and f) Examine the hundreds and note the hundreds; exit the first line and begin again or add and check (RENAME). Instruction with the RENAME strategy involved verbal rehearsal until the student could look at the first letter of the mnemonic and state the strategy step. Abstract level instruction (lessons eight through ten) involved the use of numbers only and the RENAME strategy. Using the explicit instruction steps, the teacher and student solved problems with the RENAME mnemonic within sight, but no other visual or manipulative aids.
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, no instruction occurred. The students had no exposure to multiplication. The researcher administered probes, by giving a student a probe, setting timer for 2 minutes. After two minutes, the researcher took the probe, provided no feedback, but thanked the student for participating
Please describe how replication of treatment effect was demonstrated (e.g., reversal or withdrawal of intervention, across participants, across settings) replicated across students at three different points in time. Ed and Jack were in the same group.

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 multiple probe across students design was used to investigate the presence of a functional relation between CRA and SIM instruction on students’ multiplication with regrouping performance. Prior to the study, baseline stability was defined as at least five data points in which the last three did not vary more than twenty percent from the mean of the baseline data path. All students began baseline and the first student moved from baseline to instruction after demonstrating stable baseline performance. The second student began instruction after the first student reached the criterion for phase change (twentyfive correct digits) and his baseline performance was stable. The third and fourth students began instruction after the second student reached the criterion for phase change and his baseline performance was stable. In order to evaluate the presence of a functional relation, student data were inspected visually with attention to the level, trend, and overlap of data paths across students.
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
 1
 Minimum group size
 1
 Maximum group size
 2
What was the duration of the intervention (If duration differed across participants, settings, or behaviors, describe for each.)?
 Weeks
 4.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 25.00
 Weeks
 4.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 25.00
 Weeks
 4.00
 Sessions per week
 3.00
 Duration of sessions in minutes
 25.00
 What were the background, experience, training, and ongoing support of the instructors or interventionists?
 The researcher was the interventionist. The researcher had a PhD and was a university faculty, former special education teacher ( 4 years of experience), and had maintained current teacher certification.
Describe when and how fidelity of treatment information was obtained. Treatment fidelity data were collected using a checklist of teacher behaviors associated with probe administration and CRASIM instruction. Checklists were completed by an observer two out of three sessions per week (66%); therefore data were collected across all students, during baseline, and across all instructional phases (concrete, representational, and abstract levels).
What were the results on the fidelityoftreatment implementation measure? Treatment fidelity was 100% throughout the study.
Was the fidelity measure also used in baseline or comparison conditions? both
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 and metric of magnitude of change
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 data were collected across all students and data paths remained stable for all students prior to instruction. Mari’s baseline data ranged from twelve correct digits to fourteen correct digits with a level of thirteen. Jon’s baseline data ranged from ten to thirteen with a level of eleven correct digits. Ed’s baseline data ranged from seventeen to twenty with a level of 17.6 correct digits. Jack’s baseline ranged from nine to thirteen with a mean of 10.6 correct digits. All students demonstrated similar patterns of performance across their baseline probes. The students partially completed problems, but the portion completed was incorrect. Two of the students multiplied the multiplicand by the number in the ones place of the multiplier. They completed the problems as if the multiplier was a onedigit number (e.g., 22x36=132). Another error pattern was multiplying vertically; multiplying the numbers on the ones place and writing the product without regrouping then multiplying the numbers on the tens place and writing the product (e.g., 22x36=612). Intervention Mari’s performance decreased after instruction began. There was considerable overlap between baseline and instruction. However, this decrease appeared to be the result learning. Prior to instruction, Mari completed problems without hesitation and moved from problem to problem quickly. After instruction began, Mari spent more time on problems and appeared to be puzzled as evidenced by her facial expression as well as marks and erasures on the paper. Rather than ignoring the second digit in the multiplier, she appeared to attend to it, but did not know how to proceed in solving the problem. As instruction progressed from the concrete level to the abstract level, Mari’s procedural errors decreased. At the abstract level, she used appropriate procedures and increased her speed. Mari made incremental progress toward mastery with an increasing data path that ranged from ten to thirtyseven correct digits with a level of twentyone and forty percent overlap between baseline and intervention. She reached mastery after eleven probes. Mari was given a maintenance probe one week after instruction because she was supposed to move to another school; she maintained her performance. Additional maintenance data were not available and generalization data were not collected since Mari moved to another school. After instruction began, Jon’s performance remained similar to baseline while in the concrete stage of the intervention. Although it appears that his performance decreased with the third probe, his approach to problems changed; he wrote fewer correct digits and attempted only one problem, but his procedures were more correct. Jon made incremental progress toward mastery, showing an increasing data path. His probes ranged from two to thirtysix with a level of eighteen and forty percent overlap between baseline and intervention. Jon maintained his performance, writing thirtyeight correct digits two weeks after instruction ended, thirtysix correct digits three weeks after instruction ended, and fortyfour digits after four weeks of no instruction. Jon was given a generalization probe after maintenance data were collected (four weeks after instruction) which involved multiplication with a threedigit multiplicand (e.g., 245x25). Jon wrote twentysix correct digits. After instruction began, Ed’s performance remained similar to baseline while in the concrete stage of the intervention. During the representational phase of instruction, Ed’s performance began to increase and he met the criterion rather quickly. Ed had seven probes to the criterion for mastery with a level of thirty and twentynine percent overlap between baseline and intervention. Ed’s data path showed an increasing trend with a range from seventeen to fiftynine. He maintained his performance, writing fifty correct digits two weeks after instruction and fiftynine digits after three weeks of no instruction. Ed completed a generalization probe after maintenance data were collected (three weeks after instruction); Ed wrote fortyfour correct digits. After instruction, Jack’s performance decreased to levels below baseline. Jack’s performance did not increase until instruction at the abstract level began; then his data path showed an increasing trend. Jack met criterion after twelve probes with a level of sixteen and fiftyeight percent overlap between baseline and intervention. Two weeks after instruction ended, Jack maintained and increased his performance, writing fifty correct digits. Jack was given a generalization probe after maintenance data were collected (two weeks after instruction); Jack wrote thirtytwo correct digits. Effect Size TauU was calculated for all students as well as overall. This procedure combines analysis of nonoverlapping data points between phases with trend within the intervention phase while accounting for any trend within baseline (Parker, Vannest, Davis, & Sauber, 2011). There were no significant trends during baselines phases for any of the students For Mari, a moderate effect was indicated (TauU=0.6). In comparing baseline and intervention phases for Jon, a moderate effect was indicated (TauU=0.6). A comparison of Jack’s baseline and intervention phases revealed a small effect (TauU=0.1) There was a moderate effect indicated for Ed (TauU=0.7). Overall, the intervention had a moderate effect across all students (TauU= 0.5).
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
Most tools and programs evaluated by the NCII are branded products which have been submitted by the companies, organizations, or individuals that disseminate these products. These entities supply the textual information shown above, but not the ratings accompanying the text. NCII administrators and members of our Technical Review Committees have reviewed the content on this page, but NCII cannot guarantee that this information is free from error or reflective of recent changes to the product. Tools and programs have the opportunity to be updated annually or upon request.