Strategic Math Series: Standard Algorithm
Study: Flores & Hinton (2019)
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. (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.
Participants
 Describe how students were selected to participate in the study:
 In order to participate, students had to meet the following criteria: (a) parent permission to participate in research, (b) proficiency in addition and subtraction with regrouping as defined as writing at least 20 correct digits within two minutes, (c) fluency in basic multiplication as defined as writing 30 correct digits within one minute, and (d) lack of proficiency in multiplication with regrouping as defined as less than 25% of problems competed correctly on a multiplication with regrouping probe. At the point of intervention, the students had received instruction in multiplication with regrouping in their general education classroom and as part of targeted instruction, or tier two intervention. Tier two intervention for these students involved small group instruction implemented during a designated time period in which all students in the school received either enrichment or remediation. Tier two interventions involved repeated lessons using curriculum materials previously used in students’ general education instruction, but within a smaller group, over the course of a nineweek period. Tier two instruction did not result in improvement of the students’ mathematics grades and the school problem solving team determined that they had not made adequate progress based on monthly assessments. The authors were not given access to the students’ assessment records, so specific progress monitoring procedures cannot be discussed.

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 design.
Clarify and provide a detailed description of the treatment in the submitted program/intervention: The instructional materials consisted of an instructional manual with lesson guidelines and suggested scripts to guide teacher behaviors and language. The student materials consisted of ten lessons, three for concrete instruction, three for representational instruction, one lesson to teach the RENAME strategy, and three abstract lessons. Additional abstract instructional lessons were available in case students did not reach the criteria for mastery after ten lessons. Concrete lessons involved problems using short scenarios and their translation into mathematics problems. Concrete level instruction included a laminated place value mat and a set of base ten blocks. Student learning sheets for representational instruction presented problems using short scenarios and a place value table printed next to the problem. Abstract level instruction included problems written using numbers only as well as word problems that required multiplication, addition, or subtraction. In addition to demonstrating computation using numbers only, students discriminated among operations in order to correctly solve word problems. Multiple word problems were presented and taught so that students would not be encouraged to solve all word problems using multiplication. Students would use what they learned about the operation to discriminate its application from that of other operations. Instructional procedures. One of the researchers served as the instructor and worked with each student individually. The research provided instruction four days a week for 20 minutes each day for three weeks. Different lengths are attributed to variance in student progress. For example, one student mastered the skill within three weeks over the course of ten lessons, another student mastered the skill at the threeweek point with eleven lessons, and the third student mastered the skill in three and onehalf weeks with fourteen lessons. The instructor used the explicit instructional sequence throughout each level of instruction. The instructor provided: (a) an advance organizer, telling the student what would happen over the course of the lesson, (b) demonstration of problem solving through physical actions as well as thinking aloud, (c) guided problem solving, in which the instructor and student took turns trading tasks back and forth with prompting, (d) independent practice, in which the student solved problems without assistance, and (e) a post organizer, in which the instructor briefly reviewed the lesson events. Instruction began at the concrete level with the presentation of sentences that were translated from words into multiplication equations. Rather than teaching multiplication with regrouping using just equations, sentences were used to show the application of the multiplication concept, repeated addition, or adding groups that each included the same amount. The translation process involved scaffolding. An example sentence was, There are 24 boxes of books and each box has 6 books. How many books in all? The information from these sentences was used to tell how many groups and how many things were in each group. On the lesson sheet, there was a place to note this. Once the number of groups and the number of objects in each group were established, a multiplication equation was written using numbers and symbols (i.e., ___ x ___ was filled in with 24 x 6). Finally, the student and teacher solved the multiplication problem, written vertically. With the assistance of base ten blocks and a multiplication mat, the student and teacher solved the vertically written multiplication equation. Using base ten blocks, the multiplicand (i.e., top number) was decomposed (e.g., 24 was two tens and four ones) and blocks were placed in the appropriate columns on the place value mat. Using the blocks and mat, problem solving according to the traditional algorithm began. Starting with the ones, the instructor and student made groups of blocks according to the multiplier (six groups of four ones). Regrouping occurred according to the following rule, if there are ten or more, go next door. In this problem example, six groups of four resulted in 24, a number that was more than ten. Ones blocks were exchanged for tens blocks and the tens blocks were placed in the tens column of the mat. On the written problem, the number two was written above the numeral in the tens place of the multiplicand to note regrouping. Next, the number of ones blocks that remained on the mat was noted on the written problem (four blocks remained in the ones column of the mat, so the numeral four was written in the ones place of the problem). According to the traditional algorithm, the numeral in the tens place of the multiplicand is multiplied by the multiplier. To complete this process, the instructor and student made groups of tens blocks according to the multiplier (e.g., six groups of two tens). After making groups, the instructor and the student counted the number of tens and applied the rule about ten or more (the problem example involved 12 tens). Tens blocks were exchanged for a hundreds block and the hundreds block was placed in the hundreds column of the mat. The instructor and student marked the problem according to the blocks on the mat. Before moving to another problem, the instructor and student compared the place value mat with the numbers in the written problem. If the answer for the written problem was 144, the place value mat had one hundreds block, four tens blocks, and four ones blocks in the appropriate columns. Representational level instruction involved the use of the same written sentences. However, the translation of the sentence into a mathematical equation was verbal without the written prompts used at the concrete level (e.g., ___ groups of ___ and ___x___). Rather than using base ten blocks, the instructor and student drew representations of the base ten blocks. Hundreds were drawn as squares, tens were drawn as vertical lines, and ones were drawn as short tallies written on a horizontal line. The instructor and student drew on a replica of the place value mat that was printed on the student learning sheet. Beginning with the ones column of the written problem, the instructor and student drew groups of ones according to the multiplier (six groups of four ones). For this problem example, there were ten or more. Two groups of ten tallies were circled and long lines were drawn horizontally above the tens place. Regrouping was noted on the written problem by writing a small numeral above the tens place. Using the number of remaining ones (i.e., uncircled short tallies), the teacher and student wrote the number on the written problem in the ones place. According to the multiplication algorithm, after multiplying numbers in the ones place, the next step is multiplying by the numeral in the tens place of the multiplicand. The instructor and student drew groups of tens according to the multiplier (i.e., six groups of two tens). After drawing the groups, the instructor and the student counted the number of tens and applied the rule about ten or more. Since there were ten or more in this example problem, 10 long vertical lines were circled and a square representing one hundred was drawn in the hundreds column of the table. The instructor and student marked the problem according to the drawings. Before moving to another problem, the instructor and student compared the drawings with the numbers in the written problem. After representational instruction, the seventh lesson involved learning the RENAME strategy. The instructor solved a problem using RENAME, demonstrating the strategy’s use. The student recited the strategy steps as they were written. The instructor showed the first letter of each step and the student recited the step. This continued until the student could recite the strategy when given just the mnemonic device. Instruction at the abstract level began once the student could recite each step of the RENAME strategy. Abstract instruction involved computation of problems using only the strategy. Problems within abstract lessons were presented two ways: written numerals only in the form of vertically written equations and onestep word problems that required addition, subtraction, or multiplication with regrouping. The instructor and student used the RENAME strategy to solve multiplication problems. Solving word problems involved discrimination among the operations and problem solving using RENAME. For example, the instructor and student thought aloud and talked about what was happening within the word problem, whether numbers were combined to find the answer (e.g., addition or multiplication) or separated to find the answer (e.g., subtraction). If the numbers were combined to find the answer, the teacher and the student determined whether there were groups, each with the same amount (e.g., multiplication) or whether there were groups with different amounts combined (e.g., addition). Since the students were proficient in addition and subtraction prior to the study, computational procedures using RENAME for addition and subtraction were not included in instruction. Once the teacher and student correctly identified the operation needed to solve word problems, students solved addition and subtraction problems without the instructor’s assistance. Multiplication problems were solved using the RENAME strategy. Problems involving division were not included because, at the beginning of the study, division instruction had not been completed within their general education mathematics classroom. As stated in the materials section, there were ten lessons total and extra lessons were included at the abstract level. Toni completed the intervention and reached criteria after ten lessons. Tom completed ten lessons plus one extra lesson at the abstract level. Tina completed ten lessons plus four additional abstract lessons before reaching the criteria for mastery.
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]): No instruction occurred in baseline. The researchers administered probes. The research gave the student a probe, started a timer for 2 minutes. At the end of 2 minutes, the researcher took the probe, provided no feedback and thanked the student for working hard. There was no exposure to multiplication during baseline.
Please describe how replication of treatment effect was demonstrated (e.g., reversal or withdrawal of intervention, across participants, across settings) 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 effect at three different points in time. The researchers investigated the relation between CRASIM and regrouping performance using a multiple probe across students design. Baseline data were collected until the data path was stable, as defined as the last four data points prior to intervention having no more than 20% variation from their mean. The first student moved from baseline to intervention when baseline data were stable and continued until the criteria for mastery were reached, writing at least 30 correct digits within twominutes with 100% of problems completed correct, meaning that the answers were correct. The criteria included 30 correct digits because that is the fluency standard for third grade students (Hosp, Hosp, & Howell, 2007). The second student began intervention when the first student wrote at least 20 correct digits within twominutes and 100% of the problems completed were correct. This student continued the intervention until the criteria for mastery were met. The third student moved from baseline to intervention when the baseline data path was stable, and the second student wrote at least 20 correct digits within twominutes and 100% of the problems completed were correct. The third student continued in the intervention until the criteria for mastery were met. After the criteria for computation were met, each student completed one untimed problemsolving probe and an interview regarding their computation procedures. The researchers gave a maintenance computation probe two weeks after the study.
If the study is a multiple baseline, is it concurrent or nonconcurrent? Concurrent
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
 1
What was the duration of the intervention (If duration differed across participants, settings, or behaviors, describe for each.)?
 Weeks
 3.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 20.00
 Weeks
 3.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 20.00
 Weeks
 3.00
 Sessions per week
 4.00
 Duration of sessions in minutes
 20.00
 What were the background, experience, training, and ongoing support of the instructors or interventionists?
 The researchers were the interventionists. They were university faculty with PhDs, former special education teachers (48 years of experience) who still held their teaching certificates.
Describe when and how fidelity of treatment information was obtained. The researchers measured treatment fidelity through direct observations and completion of a checklist of instructor behaviors. Treatment fidelity was completed three out of four days each week (i.e., 75% of the lessons) exceeding recommendations for high quality single case design (Poling, Methot, & LaSage, 1995). The second author and another trained observer completed the checklist and indicated whether behaviors were observed or not observed.
What were the results on the fidelityoftreatment implementation measure? Treatment integrity was calculated as 100% with 100% interobserver agreement.
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 a 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. Toni Toni’s mean score for the baseline phase or level was 8.5 correct digits (SD = 1.0). Zero percent of the completed computation problems were correct, meaning none of the answers to the equations computed were correct. Both data paths (correct digits and percent correct) were stable in trend. There was an immediate change in the number of digits written between baseline and intervention, but no change in the percentage of problems correct upon phase change. For the intervention phase, the level (mean) of the data path for correct digits was 31.4 with data points ranging from 19 to 44 (SD = 10.76). The level (mean) of the data path for percentage of completed problems correct was 69% with data points ranging from zero to 100%. There were nine probes completed before Toni met the criteria for mastery (i.e., 30 correct digits with 100% of problems completed correct). The PND for correct digits and percent correct were 0% and 20% respectively. Two weeks after instruction ended, Toni demonstrated maintenance by writing 43 correct digits with 100% of the answers correct. Tom Tom’s mean score for the baseline phase or level was 3.1 correct digits (SD = 2.97). He completed 0% of computation problems correctly. Both data paths (correct digits and percent correct) were stable in trend. There was an immediate change in the number of digits written between baseline and intervention, but no change in the percentage of problems correct at phase change. The level (mean) of the data path for correct digits was 23 with data points ranging from 9 to 37 (SD = 10.59). The level (mean) of the data path for percentage of completed problems correct was 59% with data points ranging from zero to 100%. There were ten probes completed before Tom met the criteria for mastery (30 correct digits with 100% of problems completed correct. The PND for correct digits and percent correct were 0% and 30% respectively. Two weeks after instruction ended, Tom maintained his performance by writing 33 correct digits with 100% accuracy. Tina Tina’s mean performance during the baseline phase (level) was 3.4 correct digits (SD = 2.95). She completed 0% of computation problems correctly. Both data paths (correct digits and percent correct) were stable in trend. At phase change from baseline to intervention, Tina’s data were similar; the correct digits increased slowly, and the percentage correct remained the same until there was a large increase at the fourth data point in the path. The level (mean) of the data path for correct digits was 30.7 with data points ranging from 7 to 48 (SD = 13.46). The level (mean) of the data path for percentage of completed problems correct was 68% with data points ranging from zero to 100%. There were 13 probes completed before Tina met the criteria for mastery (i.e., 30 correct digits with 100% of problems completed correct). The PND for correct digits and percent correct were 8% and 23% respectively. Two weeks after instruction ended, Tina maintained her performance by writing 48 correct digits with 100% accuracy. Effect Size The researchers calculated TauU for each student. There were no significant trends for any of the students within baseline phases. In comparing Toni’s baseline and intervention phases for correct digits, a strong effect was indicated (TauU=1.0). For Tom, there was a strong effect indicated for correct digits between baseline and intervention phases (TauU= 1.0). In comparing Tina’s baseline and intervention digits correct data, a strong effect was indicated (TauU=0.98). The researchers found an overall effect for the study with regard to correct digits, finding a strong effect across all students (TauU=0.99). With regard to effect size calculations for the percentage of completed problems correct, the effects were moderately strong. The comparison between baseline and intervention phases for percentage of answers correct, a moderately strong effect was indicated for Toni (TauU=0.78), Tom (TauU=0.70), and Tina (TauU=0.77). The researchers found the overall effect with regard to percentage of problems correct to be moderately strong (TauU=0.75).
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.