Strategic Math Series: Partial Products
Study: Flores & Milton (2020)

Describe how students were selected to participate in the study:

The criteria for participation were as follows: a) mastery of multiplication facts with digits 0–5, b) deficit in multiplication with regrouping as defined as writing less than 20 correct digits in one minute and less than 50% accuracy in obtaining correct products, c) eligibility for special education services with individualized education program goals related to multiplication of multi-digit numbers, and d) parent permission for participation.

Describe how students were identified as being at risk for academic failure (AI) or as having emotional/behavioral difficulties (BI):

Two students qualified for special education services under the category of other health impairment because they had a medical condition of Attention Deficit Hyperactivity Disorder that affected attention and learning. The third student qualified for special education services under the category of specific learning disability (SLD). The eligibility criteria for SLD was a 16-point or greater difference between predicted and actual achievement. Predicted achievement was obtained using the student’s cognitive ability score and a regression table.

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:

The researchers implemented a multiple probe across students design (Horner & Baer, 1978). Each of the students began in baseline. The first student moved from baseline to intervention when baseline data were stable as defined as the last three data points varying no more than 20% from the mean of baseline. Once the first student demonstrated at least 60% accuracy across three probes, the second student moved from baseline to intervention. The third student moved from baseline to intervention when the second student showed an improvement of three probes at or above 60% accuracy. The researchers defined mastery as at least three consecutive probes with at least 30 digits written correctly and all completed problems (100%) had correct products. The researchers collected fluency and accuracy data for each probe because it with possible to write many correct digits across the partial products but fail to arrive at the correct product. Therefore, the researchers intended to ensure that students demonstrated fluency in completing problems correctly. After students completed the intervention, they completed a maintenance probe each week for up to eight weeks. One year after the intervention, the researchers administered another maintenance probe to participants.

Clarify and provide a detailed description of the treatment in the submitted program/intervention:

Instructional Procedures Lesson structure. The teacher used explicit instruction during each lesson. This included an advance organizer in which the teacher previewed the lesson and provided expectations for behavior. Next, the teacher modeled by physically showing the problem-solving process and thinking aloud. She involved the students in the process by asking them to repeat information, count with her aloud, or provide answers to known mathematics facts. The third step was guided practice. The teacher and students solved problems in a back and forth process with the teacher completing one part of the problem and the student completing the next part. The teacher provided prompts when needed. After guided practice, the teacher conducted independent practice by giving students problems to complete without her assistance. The last step was a post organizer in which the teacher highlighted the important parts of the lesson and previewed the next lesson. Concrete instruction. The first four lessons were concrete and included the use of base ten blocks, a place value mat, and learning sheet. The teacher read the word problem and talked about what was happening. For example, multiple shelves each had the same number of books and they were finding how many in all by combining groups with the same amount in each. Completed an item about groups (e.g., ___ of ___) by using the problem parts to complete the phrase (e.g., 24 groups 23). Then, the teacher talked about translating that phrase into a multiplication equation and wrote the equation. The teacher implemented the following steps to compute the problem, 24 x 23. She talked about the composition of each number: twenty-three included twenty and three and twenty-four had twenty and four. She told the students that she would multiply each of the parts to make partial products. She explained that the word, partial, meant part. She explained that she would make equations with the digit in the ones place of the multiplier. She talked about the commutative property and that she could make four groups of three or three groups of four. She made equal groups on the multiplication mat in the ones place. The answer was 12, so the teacher showed regrouping by exchanging 10 ones for a tens block and placing them on the mat in the appropriate place value column. She wrote 12 in the problem and talked about where to write the numbers, emphasizing that there was one ten in the tens place and two ones in the ones place. On the mat, the next set of rows was shaded gray. She made the equation to find the second partial product with the multipliers 20 and three. She made three groups of two tens blocks (3 x 20). The teacher talked about the product shown with blocks on the mat and wrote 60 in the problem. She referenced the blocks on the mat and talked about how to write 60 with six in the tens place and zero in the ones place. The teacher continued making equations with blocks and writing the other partial products in the problem (80 and 400). The last step was adding all the partial products to find the total product. Beginning with blocks in the ones column and moving to the tens and hundreds, the teacher moved all the blocks in each column to the bottom of the mat. She wrote the sums from each column in the written problem. The teacher checked the answer for reasonableness. She estimated each of the multipliers by rounding to the nearest ten or a familiar number (e.g., 20 x 25), and she multiplied the estimates. She talked about making an easier problem that she could complete mentally to check her work. The estimated answer was 500 and that was close to the actual answer; it was reasonable. Students could not move to the next set of lessons unless the student achieved 100% accuracy on the independent practice problems within the concrete lessons. Figure 2 shows a problem represented on the place value mat using base ten blocks. Representational instruction. Lessons five through eight were representational and involved drawing problems on a place value table printed on the learning sheet. Solving problems involved the same steps as concrete instruction. The teacher and students drew ones as short tallies, tens as long vertical lines, and hundreds as squares. To show regrouping, they circled the drawings. Students could not move to the next set of lessons unless they achieved 100% accuracy on representational lessons. Abstract instruction. The teacher taught the students a mnemonic strategy to help them remember the procedural steps of the partial products algorithm. In lesson nine, students memorized the strategy before using it to solve problems. The strategy was as follows: (a) Read the problem; (b) Examine the ones in the multiplier to make equations; (c) Note the partial products for ones; (d) Address the tens in the multiplier to make equations; (e) Mark the partial products for tens; and (f) Examine the columns, add, and check (RENAME). It is important to note that the second step prompted students to examine the ones in the multiplier and think about the two equations that would be made. The third step of the strategy prompted students to execute the operations and write the two partial products generated from those equations in the problem. The fourth and fifth steps prompted similar responses but using the tens place of the multiplier. The processes associated with the RENAME strategy are shown in Figure 3. There was one lesson in which the students applied RENAME to multiplication equations. In order to prepare students for discrimination among operations, the teacher taught students an additional strategy. The strategy was as follows: (a) Find what you are solving for; (b) Ask yourself, “What are the parts of the problem;” (c) Set up the numbers; and (d) Tie down the sign. The students learned to find question within the word problem. The next step was thinking about that was happening in the problem by finding the parts of the problem. The teacher taught the students to notice whether objects came together or separated. If objects came together, they noticed whether groups of the same size combined (multiplication) or groups of different sizes combined (addition). The students recognized subtraction as separation of a group. Then, they used the parts of the problem to identify the numbers that would be used, they confirmed the operation and wrote the equation. The students completed lessons in which they solved word problems that required addition, subtraction, or multiplication. When the problems involved multiplication, they used the two strategies, FAST RENAME. They also completed equations using RENAME.

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, students received no instruction related to multiplication. The teacher administered probes (same as those used in intervention phases). The probes were timed for two minutes. After one minute, the teacher provided no feedback and thanked the student for participating. Each student worked with the teacher one-on-one. Students in baseline had no exposure to the intervention because their scheduled instruction occurred at different times of the day.

Please describe how replication of treatment effect was demonstrated (e.g., reversal or withdrawal of intervention, across participants, across settings)

Multiple probe 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 of effect at three different points in time. The decision to move from baseline to treatment was response-based. For the first student, treatment began after stable baseline (stable baseline, last three data points varying no more than 20% from the mean of baseline). When student one reached 60% accuracy cross three probes, student two began intervention if baseline data were stable. The same procedures occurred for student three. The criteria for mastery of at least 3 consecutive probes with 30 correct digits and 100% accuracy in products. Each student, at a different point in time reached this mastery criterion.

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: