Strategic Math Series: Standard Algorithm
Study: Flores & Hinton (2019)

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 nine-week 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,
• 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 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 three-week point with eleven lessons, and the third student mastered the skill in three- and one-half 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., un-circled 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 one-step 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., 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]):

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 CRA-SIM 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 two-minutes 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 two-minutes 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 two-minutes 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 problem-solving 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 non-concurrent?

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: