Strategic Math Series: Standard Algorithm
Study: Flores, Hinton, & Schweck (2014)

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 one-digit multiplier; and d) inability to compute multiplication with regrouping problems that included a two-digit multiplier. All of the students met these criteria and demonstrated similar deficits in their ability to complete problems with two-digit 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,
• non-responsive to Tiers 1 and 2, or
• designation of severe problem behaviors on a validated scale or through observation?

%

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., multi-element 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 (twenty-five 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 non-concurrent?

Non-concurrent

How was the program delivered?

Individually
Small Group
Classroom

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 word-attack skills, a targeted measure would be decoding of pseudo words. If a program taught comprehension of cause-effect passages, a targeted measure would be answering questions about cause-effect 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 word-level 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 on-task behavior in one classroom, a broader measure would be on-task 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.

If you have excluded a variable or data that are reported in the study being submitted, explain the rationale for exclusion: