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Schema-based strategy instruction and the mathematical problem-solving performance of two students with emotional or behavioral disorders.

Abstract

The purpose of this study was to analyze the effects of schema instruction on the mathematical problem solving of students with emotional or behavioral disorders (EBD). The participants were two fourth-grade students identified with EBD. The intervention package consisted of schema instruction, strategy instruction on problem-solving heuristics (i.e., understand the problem, make a plan, carry out the plan, check), and the use of reinforcement for task completion. The study used a single case experimental design (SCED) to establish a functional relation between the intervention package and problem-solving performance. A functional relation was established and results demonstrated an improvement in problem solving on addition/subtraction and multiplication/division word problems. The Tau-U effect size was used to corroborate visual analysis results. The Tau-U for participants 1 and 2 was 100% and 96%, respectively. The students and special education teacher reported that schema instruction was a socially valid intervention.

Keywords: emotional and behavioral disorders, mathematics, problem solving, schema instruction, single case experimental design

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Students with emotional and behavioral disorders (EBD) often struggle with mathematics (Alter, Brown, & Pyle, 2011), frequently fail competency exams, and perform poorly compared to same-aged peers with and without disabilities (Nelson, Benner, Lane, & Smith, 2004). A comprehensive literature review on the mathematical performance of students with EBD found these deficits worsened across middle and high school in 12 of 13 studies (Trout, Nordness, Pierce, & Epstein, 2003). Mathematical problem solving is emphasized in current mathematics standards; however, the dearth of empirical evidence for problem-solving interventions for students with EBD is concerning.

In addition to the limited number of studies, there are two weaknesses in the existing EBD academic intervention literature targeting mathematics. First, existing intervention research tends to use lower-level skills as dependent variables to illustrate effects. Second, the intervention work often addresses regulatory behaviors associated with academic performance rather than skill or strategy acquisition related to the academic content. Support for the first limitation is found in several recent meta-analyses of interventions in mathematics for students with EBD. These meta-analyses report a majority of the research focuses on basic level skills such as basic facts and computation and below grade level concepts (Hodge, Riccomini, Buford, & Herbst, 2006; Mulcahy, Maccini, Wright, & Miller, 2014; Ralston, Benner, Tsai, Riccomini, & Nelson, 2014; Templeton, Neel, & Blood, 2008). These findings indicate a need to establish intervention effects with more complex mathematical concepts such as problem solving and upper grade level content. Lane, Barton-Arwood, Nelson, and Wehby (2008) addressed the second limitation by stating the focus of existing intervention literature for mathematics and students with EBD targets memory and/or self-regulatory interventions.

There is a sample of five experimental studies targeting the problem solving of students with EBD. Four author teams conducted these studies and included 16 students (Alter et al., 2011; Alter, 2012; Maccini & Ruhl, 2000; Mulcahy & Krezmien, 2009; Swanson, 1985). All five studies used a single case experimental design (SCED) to establish a functional relation between the intervention package and problem-solving performance; however, none of the studies reported effect sizes alongside visual analysis. Four studies with 13 participants found heuristic instruction (i.e., understand, plan, solve, and check) effective in increasing students' problem-solving performance. The fifth study included three students and found self-instruction effective in improving problem solving. Jitendra, George, Sood, and Price (2009) implemented schema instruction with two students and reported pre- to post-test gains of 27-97% and 73-98%; however, the study did not use an experimental design.

Schema Instruction Literature

A thorough literature base has shown schema instruction is effective in increasing students' problem-solving performance (Jitendra et al., 2015); however, application to the EBD population has not been experimentally tested. The term schema originates from psychological and philosophical theory. Schema is defined as a framework developed to solve a problem (Marshall, 1995). Schema instruction involves the identification and classification of problems into types by analyzing the underlying structure. With this structure known, students are able to identify an appropriate solution plan.

Several randomized control trials that implemented schema instruction reported positive effects for students with and without learning disabilities (Fuchs, Fuchs, Prentice, et al., 2004; Fuchs et al., 2008, 2009, 2010; Jitendra, Dupuis, et al., 2013). Powell (2011) conducted a literature review and reported that 12 group design studies implementing schema instruction yielded positive effects on the problem-solving performance of students with and without learning disabilities. However, an aggregated effect size was not reported. Schema instruction has been classified into two approaches: schema-based instruction and schema-broadening instruction (Powell, 2011). Schema-based instruction uses schematic diagrams in addition to typical schema instruction (Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007; Jitendra et al., 2009; Jitendra, Dupuis, et al., 2013; Jitendra, Rodriguez, et al., 2013). Schema-broadening instruction teaches students to identify "transfer features" to enhance students' ability to solve novel problems (e.g., Fuchs et al., 2003, 2008; Fuchs, Fuchs, Finelli, et al., 2004). Another component consistent across the schema literature (e.g., Fuchs et al., 2008, 2010; Jitendra et al., 2007; Jitendra, Rodriguez, et al., 2013) is the use of a mnemonic to prompt the four critical steps of problem solving: (1) understand the problem, (2) devise a plan, (3) carry out the plan, and (4) look back (Polya, 1945).

Current Study

The purpose of this SCED was to examine the effect of schema instruction on the problem-solving performance of two students with EBD. The intervention package included schema instruction, strategy instruction on a problem solving heuristic (i.e., Search the problem, Translate the problem into a schematic diagram, Answer the problem, Review the solution; Maccini & Hughes, 2000; Maccini & Ruhl, 2000), and reinforcement for task completion. Explicit instruction was used to provide schema instruction, the use of schematic diagrams, and the use of a problem-solving mnemonic (e.g., Jitendra et al., 2007, 2009; Jitendra, Dupuis, et al., 2013). A SCED was used to answer research questions about the effects of a schema instruction package on: (a) accuracy in representation of the schematic structure of addition/ subtraction (part-part-whole, change, compare) and multiplication/ division word problems (equal groups) using schematic diagrams; (b) accuracy in solution to word problems; and (c) social validity of the instructional package for students and teacher.

Method

Setting and Participants

The study was conducted in an elementary school in a large town in the south-central United States. The school enrolled 465 students in pre-kindergarten through fifth grade. The population was diverse (69% Hispanic, 20% African-American, 10% Caucasian). In addition, 30% of students were English second language learners and 93% of students received free or reduced meals. One self-contained classroom for students with EBD functioned on campus. Six students (4 second-graders and 2 fourth-graders) received instruction in this setting as dictated by their Individualized Education Programs (IEP). A seventh student transferred out of the school during the first week of the study. This was confirmed through direct observation.

Prior to the start of the study, the special education teacher and paraeducator provided all instruction to the students. The teacher was a Caucasian male and the paraeducator was a Hispanic female. The teacher had two years of special education teaching experience, and this was the paraeducator's first year of teaching. The teacher was dual certified to teach special education and elementary general education; the paraeducator's certifications were not known. A pre-service teacher was present in the classroom two times per week before the start of baseline and throughout the study. The pre-service teacher observed and provided instructional support.

To recruit students for the study, the first author communicated with the special education teacher, who nominated students meeting the following criteria: identified with EBD, dedicated instructional time in mathematics, and deficits in problem solving. The teacher suggested including two fourth-grade students who met the inclusion criteria (i.e., identified with EBD, deficits in problem solving, received dedicated instructional time in mathematics). The Institutional Review Board (IRB) for the affiliated university and school district granted permission. Students granted assent; legal guardians provided consent. The study was not blind to participants, parents, or teacher.

Three separate observations were conducted across three weeks. Observations confirmed dedicated instructional time in mathematics was provided for the two potential students. Observations indicated no schema or strategy instruction on a problem-solving heuristic was provided by the teacher or used by students. Typical instruction consisted of whole-group and individualized instruction; group sizes ranged from one-on-one to six students in a group depending on the objective for the day. Guided and independent practice included the use of one-on-one assistance provided by the classroom teacher, para-educator, or pre-service teacher. Problem-solving instruction included the use of manipulatives, pictorial representations, and key word strategies. Classroom management incorporated a token economy to encourage appropriate academic behaviors. An informal interview with the teacher corroborated direct observation findings that students were receiving mathematics instruction; however, there was no schema instruction provided by the teacher or used by students.

The first author (viz., the interventionist) assumed instructional responsibility at the onset of baseline to control for a Hawthorne effect during intervention. The interventionist had two years of teaching experience in an elementary school and a special education teaching certificate for kindergarten through twelfth grade. The interventionist served as the primary data collector. A non-author independently assessed data collection, scoring of measures, and fidelity of implementation through interobserver reliability and fidelity checks. The intervention baseline approximated classroom baseline conditions while controlling for treatment exposure. Probes were administered at each child's study carrel located on the side of the classroom. During intervention and probe sessions, the other students in the classroom participated in small group instruction or independent work at their desk.

Experimental Design

A SCED was used to answer questions about the effects of an intervention on a corresponding change in behavior of participants (Barlow & Hersen, 1984; Best & Kahn, 1998; Gall, Gall, & Borg, 1996). Horner and colleagues (2005) stated SCEDs have the rigor to demonstrate causal relationships and should be considered as evidence for establishing evidence-based practices. A reversal design was used to establish a functional relation between the schema intervention package and students' problem-solving performance. Experimental control was demonstrated by introducing a variable (i.e., schema instruction) to elicit a desired behavior, and when the variable was removed, the desired behavior was lost (Baer, Wolf, & Risley, 1968). To establish a functional relation, the field has recommended demonstrating at least three changes in the behavior (Kratochwill et al., 2010).

The intervention sequence occurred in phases: baseline for addition/subtraction word problems, schema-identification instruction, schema instruction on part-part-whole, schema instruction on change, and schema instruction on compare. Baseline was reintroduced for multiplication/division word problems followed by schema instruction on equal group. A maintenance phase followed the intervention.

Variables

The operational definition of mathematical problem solving included accuracy in schematic diagrams and solution of word problems involving addition/subtraction or multiplication/division. The independent variable included an academic intervention using schema instruction and strategy instruction on a problem-solving heuristic. Reinforcement was provided for task completion during baseline, intervention, and maintenance phases.

Measures

Problem solving was measured via probes containing word problems. Text complexity and numerals used in the problems were held constant across probes to control for variations in difficulty. All probe sheets were researcher created. Word problems included contextual information relevant to students' prior knowledge (e.g., using student and teacher names, the students' hometown, popular athletes). To collect face validity of the probes, the special education teacher reviewed approximately 25% of the probe sheets (i.e., aligned with grade level mathematics standard and readability). Face validity was 100%; each word problem reviewed was a match between the instructional level on the student's IEP goals and grade level content. The study used 28 probes, all available from the first author. Table 1 provides a sample probe.

Students were administered nine different probe types during the study. The first baseline probes contained two problems per schema type (i.e., part-part-whole, change, compare), six problems total. Scoring consisted of one point for correct schematic diagram and one point for correct solution. Schema-identification probes contained two problems per schema type, six problems total. Scoring consisted of one point for correct schema identification. Part-part-whole, change, and compare probes were similar. Each probe contained three problems that fit the target schema. Scoring consisted of one point for correct schematic diagram and one point for correct solution. Mixed probes contained one problem for each schema, three problems total. Scoring consisted of one point for correct schematic diagram and one point for correct solution. The second baseline, equal groups, and maintenance probes were similar; they contained three equal group problems. Scoring consisted of one point for correct schematic diagram and one point for correct solution.

Procedures

The interventionist administered all probes (i.e., baseline, intervention, maintenance) in a one-on-one setting at the child's desk located in a study carrel. The interventionist offered to read the word problems aloud. If prompted, the interventionist re-read the word problem verbatim.

A token economy was used across all probes (i.e., baseline, intervention, maintenance). A token economy connected secondary reinforcement (i.e., a check mark) to problem "attempt" within each probe sheet, controlling the opportunity for reinforcement. For every question the student attempted he received a check mark and if he earned a check for each question on the sheet, he gained access to reinforcement (i.e., edible food items). The reinforcer was identified as desirable through conversations with the students. Items met the approval of the special education teacher and fell within district guidelines.

Baseline phase. The interventionist presented the baseline probe and offered to read each problem. No time limit was given; most students completed the task in 10-15 min. A check was given for each problem attempted; if the student received a check for each problem then the reinforcer was earned. No feedback regarding correct or incorrect answers was provided.

Schema-identification instruction. The interventionist provided explicit instruction on the identification of unique features (see Table 2) for each schema type (i.e., part-part-whole, change, compare). Three schematic diagrams and worked examples for each problem type were provided. The interventionist engaged in a think-aloud to model the use of the strategy. Modeling involved the following steps: (a) read the problem aloud, (b) pose the three guiding questions (is there a whole value with different parts, is there a value changing over time, or are there two values being compared to one another), and (c) characteristics of the problem were discussed to identify the problem type.

Guided practice consisted of the presentation of a word problem to the student with three schematic diagrams (i.e., part-part-whole, change, compare). The interventionist offered to read the problem aloud. The student asked himself, prompted if not initiated, the three questions to identify the schema type. If the student provided an answer before considering the three questions the interventionist asked, "What three questions do you need to ask yourself?" The student classified the word problem, incorrect answers were followed by corrective feedback, and another problem was presented.

For independent practice, the student sorted a stack of 10 sentence-strips (12 in x 3 in) containing a word problem into the corresponding schema type. Corrective feedback was provided for each incorrect response. Once 100% accuracy was reached, the student was presented with the schema-identification probe.

Strategy instruction. Intervention sessions began with a brief discussion of the purpose and rationale for problem solving using relatable and age-appropriate contextual examples. Students were presented with a graphic organizer that listed the steps of STAR (see Table 3). The objective for the day was stated (see Table 4) followed by modeling of two to four problems.

First, the interventionist "Searched the problem" by reading the problem aloud and underlining important information (i.e., values, labels, the question; not "key words") and then placed a check next to the S to self-evaluate strategy use. Next, the interventionist "Translated the problem into a schematic diagram." In order to translate the problem into a schematic diagram the schema type was identified by asking the following questions: is there a whole with parts; is there something changing over time; or is the problem comparing two values? The corresponding schematic diagram was drawn and filled in with the underlined information; a question mark was used for the unknown value. The interventionist placed a check next to the T to self-evaluate strategy use. Next, the interventionist "Answered the question" by identifying an appropriate solution plan and performing the computation. The solution was labeled by referring back to the question that was underlined and a check was placed next to the A to self-evaluate strategy use. Finally, the interventionist "Reviewed the solution" by asking, "Is this answer reasonable?" A think-aloud discussing the reasonableness of the answer followed and a check was placed next to the R to self-evaluate strategy use.

During guided practice, the student was presented two to three word problems. If the student was unable to initiate the task, the interventionist asked the question, "What does the S stand for in our problem-solving strategy STAR?" Prompts were provided as needed to enable the student to engage in the problem-solving heuristic. At the completion of each step (i.e., S, T, A, R), the student placed a check to self-evaluate strategy use.

During guided practice, the interventionist formally assessed the student's ability to engage in the problem-solving heuristic. Additional guided practice opportunities were presented if the student made multiple errors.

During independent practice, the interventionist presented a word problem to the student and asked him to solve the problem. The student could request to have the problem read aloud. No prompts were provided while the student worked through the problem. After the completion of the problem, the interventionist analyzed the schematic diagram and solution. Corrective feedback was provided if errors were present. Additional practice opportunities were provided if errors were identified. Explicit instruction on the use of STAR was only provided on the first day of part-part-whole instruction; students used the strategy throughout the remaining sessions.

Duration. Instructional lesson duration varied. Lessons lasted approximately 20 min on the first day of instruction for a problem type and were reduced to approximately 15 min on the third day of instruction. Total instructional duration was approximately 300 min.

Maintenance. The interventionist presented the probe. The student could request to have the problem read aloud. No instruction or feedback was provided.

Data Analysis Procedures

Visual analysis of mean level, trend, stability, consistency, and intercept gap (Horner et al., 2005; Tawney & Gast, 1985) is reported alongside effect sizes with confidence intervals (Cl) to identify the presence of an effect. Interpretation of the effect size (ES) is contextualized with the visual analysis and compared to related literature (Vannest & Ninci, 2015).

Effect size. The Tau-U ES is a combination of Mann-Whitney U and Kendall's Tau (Parker, Vannest, Davis, & Sauber, 2011) and is consistent with the nonoverlap approach of dominance statistics using pairwise score comparisons (Huberty & Lowman, 2000). In the context of SCED, dominance is defined as the probability that a randomly selected datum from one phase will exceed that from another phase (Acion, Peterson, Temple, & Arndt, 2006; D'Agostino, Campbell, & Greenhouse, 2006; Parker, Vannest, & Davis, 2011). Tau-U is more robust than other nonoverlap statistical methods because of its ability to account for undesirable baseline trend and take into account data variability (Parker, Vannest, Davis, & Sauber, 2011).

The Tau-U ES was chosen as an acceptable non-parametric ES (Kratochwill et al., 2010) to report alongside SCED results. Tau-U is based on the "S" distribution, which has 91-95% of the power of parametric tests such as f-tests or ordinary least squares regressions when data are "ideal" (i.e., normally distributed and constant variance). When data are skewed and non-normal, power exceeds 100%, a typical scenario in the context of SCED. Tau-U reflects the percentage of data improvement drawing from Kendall's and Mann-Whitney U tests. To calculate Tau-U by hand, the number of improved pairs or "S" (after adjusting for monotonic trend) divided by the total number of possible pairs equals the product of the number of data points in phase A and the number of data points in phase B. Tau-U can also be calculated using an online calculator such as singlecaseresearch.org (Vannest, Parker, & Gonen, 2011).

The schema-identification, part-part-whole, change, compare, and mixed probes were aggregated into one phase of data notated as phase B. Analysis of this data compared baseline condition for addition/ subtraction word problems (A) against phase (B) intervention phase data. The second baseline condition for multiplication/division word problems was compared against the equal groups phase. Two Tau-U values were calculated for each participant and aggregated to report one Tau-U value. The participants' Tau-U values were aggregated to report a grand ES for the study. To gauge the likelihood of the reported ES, 90% CIs were reported. Aggregation of Tau-U weights each subject's data and error by using the inverse of the variance. This calculation takes into account the variability of each student's data and the amount of data when providing weights to the calculation.

Social validity. Prior to maintenance, the students and special education teacher completed a social validity survey. Questions were written in a Likert-type scale, which ranged from 1--strongly disagree to 5--strongly agree. On the student survey, emoticons were used in replace of words to depict varying degrees of satisfaction (adapted from Butler, Miller, Crehan, Babbitt, & Pierce, 2003). Each question was read to students and student questions were answered to attain valid responses. The teacher survey included an option for open response.

Interrater reliability. An undergraduate student in special education was provided with an answer key and trained to score each probe type. Reliability checks were taken on 63% of baseline and 65% of intervention probes. Interobserver agreement was calculated by adding agreements and dividing by the total number of opportunities. Agreement was 97% for baseline and 96% for intervention. All disagreements were discussed until a consensus was reached.

Fidelity of implementation. Using a checklist adapted from Alter and colleagues (2011), an undergraduate student in special education conducted fidelity observations. Training included examples and non-examples of the instructional components. Fidelity of implementation was taken on 5 out of 40 intervention sessions. Overall fidelity of implementation was 93%, with a range of 89-100%.

Results

This SCED included two participants with EBD and examined the effects of a schema instruction package on their problem-solving performance. Overall study results are discussed followed by individual results. Visual analysis involved the examination of mean level change, data stability, data overlap, trend, and intercept gap, interpreted as a functional relation and causal effect. ES quantify differences in phase change resulting from the intervention. Tau-U ES with 90% CIs are reported to corroborate visual analyses and to promote future meta-analytic work. Figure 1 contains the graphed results for each participant.

[FIGURE 1 OMITTED]

Overall

Visual analysis supports a functional relation between the intervention package and problem-solving performance. An increase in accuracy of problem solving is supported through the immediacy and consistent change with the onset of intervention for both participants.

Five baseline data probes were taken for each student; the data was relatively stable at the onset of intervention. The behavior reverted to low levels of performance during the second baseline phase, with both students answering 0% correct on three consecutive probes. The aggregated Tau-U ES for the study was 99%, interpreted as 99% of the treatment data were improved from the trend-corrected baseline data and 90% confidence that the true score is between a 70% and 100% improvement. This ES and the individual ES are in agreement with the visual analysis.

Student 1

During the first baseline, the data demonstrated some variability in the bottom quartile of performance; the data ranged from 8% to 25%. This variability is within limits of stability suggested by Tawney and Gast (1985). During the second baseline, the data was 100% stable with all data at 0% accuracy. The mean level change between the first baseline and intervention phase was 54%. The mean of the first baseline was 17% and the intervention data (i.e., schema identification, part-part-whole, change, compare, mixed) was 71%. The mean level change for the second baseline and intervention phase was 78%. The mean of the second baseline was 0% and the intervention data (i.e., equal group) was 78%. Intervention data overlapped with baseline data in one instance. The participant reached 80% proficiency for each probe type except mixed probes; the max proficiency was 68%. The Tau-U for Student 1 was 100%, which can be interpreted as 100% of the treatment data was improved from the corrected baseline data. The 90% CIs reflects the variability within the data CI equals a range of 60-100% as equally likely effect if given an unlimited sample of data.

Student 2

During the first baseline, the data demonstrated some variability in the bottom quartile of performance; the data ranged from 8% to 25%. The variability falls within the limits suggested by Tawney and Gast (1985). During the second baseline, the data was 100% stable with all data at 0% accuracy. The mean level change between the first baseline and intervention phase was 66%. The mean of the first baseline was 17% and the mean of the intervention data (i.e., schema identification, part-part-whole, change, compare, mixed) was 83%. The mean level change for the second baseline and intervention phase was 83%. The mean of the second baseline was 0% and the intervention data (i.e., equal groups) was 83%. There were no instances of overlap between baseline and intervention data. The student reached 80% proficiency for each probe type. The Tau-U for Student 2 was 97%, which can be interpreted as 97% of the treatment data was improved from the corrected baseline data. The 90% CI reflects the variability within the data CI equals a range of 56-100% as equally likely effect if given an unlimited sample of data.

Social Validity

The teacher reported he strongly agreed or agreed that the intervention: (a) targeted important behaviors, (b) was effective, and (c) could be easily implemented in his classroom (i.e., materials and time needed). The teacher stated in his open response:
   This intervention has hit on just about every area of math
   that we do as it addresses the foundations of problem solving....
   approach is concrete and simple and allowed my
   students, who are plagued with a lack of confidence, to begin
   working on attempting problems they never would
   have thought possible.


Both students self-reported that they strongly agree with the statement, "Math class is interesting." Both students self-reported that they agree or strongly agree with the statement, "I like the math activities." Both students strongly disagreed with the statement, "The math activities did not help me learn math." Finally, both students agreed or strongly agreed with the statement, "I feel confidence about mathematics."

Results from the social validity survey indicate the students and teacher felt the intervention was relevant and applicable to their current learning environment. Analysis of student data indicates pragmatically significant improvement in student problem-solving performance.

Discussion

The purpose of the current study was to test the effectiveness of a schema instruction package on the problem-solving performance of two students with EBD. A functional relation was demonstrated through a reversal design concurrently across two participants. Improvement was established from baseline to intervention with some increased variability in performance, which is a common occurrence in academic performance of students with EBD (Pierce, Reid, & Epstein, 2004). In order to analyze skill retention, maintenance data were collected following the conclusion of the intervention. Both students' performance contained variability, although it was a small sample size (i.e., three probes). The data appeared to regress for both students when comparing intervention and maintenance data. However, the mean level change from baseline to intervention was substantial for both students. Tau-U ES support claims made from visual analysis; Student 1 had an ES of 100% ([CI.sub.90] 60-100%) and Student 2 had an ES of 97% ([CI.sub.90] 60-100%). These effects are consistent with the relative effects of SCED studies implementing schema instruction with populations of students with disabilities (Rockwell, Griffin, & Jones, 2011; Xin, 2008).

During baseline, students answered approximately one or two questions correctly without understanding the underlying relationships of the quantities in the word problems. Students appeared to choose addition or subtraction at random; observing students performing computations before listening to the entire problem read aloud substantiated this hypothesis. A common error displayed by Student 1 was computation errors; the intervention did not have a computational fluency component. Jitendra, Rodriquez, et al. (2013) and Fuchs et al. (2008) embedded a component of number sense and fluency instruction in addition to schema instruction to bolster students' problem-solving performance.

The decision to not include a fluency and number sense component was made to keep the duration of each session short for implementation purposes. After the completion of the intervention, students listened to the problem, underlined relevant information, drew and completed a schematic diagram, made informed decisions on the operation to compute, and reviewed the solution for its reasonableness.

The interventionist anticipated students requiring three instructional sessions per schema to attain proficiency. Both students reached proficiency (80%) at least once per schema; however, during mixed problems the students' performance dropped. Implementing a criterion to move to the next schema (i.e., two consecutive sessions of 80% accuracy) may improve student retention. Additionally, more instructional time on identifying the schematic structure of problems was needed; observing the student choose the incorrect schema on the mixed probe sheet substantiated this claim. Both students had a propensity to ask the question, "Do we have to do this today?" The token economy was embedded only within the probe sheets and not during instruction; a prudent decision would be to embed the token economy during instruction. Students with EBD commonly display deficits in motivation (Bandura, 2006; Nelson et al., 2004); token economies embedded within mathematics interventions have been shown to increase academic behaviors associated with motivation (Alter et al., 2011; Alter, 2012).

Maintenance data for the two students was limited (i.e., three data points); however, performance was improved from baseline. The data sample was small, necessitating future research to take a larger maintenance data sample.

The social validity survey was administered post-intervention. Results from the social validity survey are significant given that student acceptance of instruction has been correlated with increased motivation and willingness to engage in academic tasks (Good & Brophy, 2008). The special education teacher reported that the intervention had a practical impact on his students and he believed the intervention package was feasible for him to implement. These findings are important because factors such as intervention complexity, time, materials needed, perceived effectiveness, and teacher motivation all influence treatment integrity and implementation (Lane, Bocian, MacMillan, & Gresham, 2004).

Implications

Prior to the intervention, typical problem-solving instruction consisted of generic problem-solving instruction embedded within the curriculum (e.g., using manipulatives, draw a picture, look for a pattern, write an equation); however, baseline performance demonstrated that the students had not demonstrated proficiency in the problem-solving process. The intervention was successful in increasing the students' problem-solving process; students utilized the STAR problem-solving strategy without prompting when completing the maintenance probes. Ultimately, students need to identify the schema of a problem, choose an operation, and solve for the solution. The mixed probes required students to engage in this process; the students only reached proficiency (80%) once on three trials. Three suggestions to increase performance would be: (a) a longer duration of instruction on each schema; (b) implementing a mastery criterion of 80% on multiple probes; and (c) spiraling instruction by providing booster sessions on schema types. Duration of each instructional session approximated 15-20 min to make application feasible under typical classroom conditions. Increasing the dosage by either (a) increasing session length or (b) providing more sessions could lead to greater student outcomes.

The on-task or engagement deficits identified for many students with EBD are a consideration in implementing strategy instruction. This study used a secondary reinforcement system or token economy as an evidence-based practice (Maggin, Chafouleaus, Goddard, & Johnson, 2011) to stabilize reinforcement across conditions, and we encourage future researchers to embed token economies in academic treatments for similar reasons.

Limitations

One limitation is the researcher served as the interventionist and primary data collector, which poses a threat to the internal validity of the study. We attempted to control for this by having an independent non-author collect interrater reliability on over 60% of baseline and intervention measures in addition to collecting fidelity checks, albeit only 12.5% of sessions. Collecting fidelity on a larger sample of sessions would have increased the internal validity. In addition, when the interventionist is the researcher, the applicability of the procedures and true social validity is less known. Social validity questionnaires were employed to address this issue. A researcher serving as the interventionist also potentially minimizes error in the procedure and may produce elevated levels of effect in the study by careful attention to detail, and higher skill level due to greater familiarity with the intervention. Future studies should include a teacher as interventionist for these three reasons.

Another limitation was the small sample (n = 2). Future researchers should aim to gather a larger potential sample of students with EBD. All instruction took place in a one-on-one setting, which may be less feasible in a general education classroom condition. Future research should analyze the effectiveness of implementing the intervention in small group or whole class applications rather than one-on-one. Another consideration for future research includes separate measures on accuracy of solution and schema representation. In this study, these two constructs were combined to take one overall accuracy score; however, interpretation is limited because solution accuracy and schema representation accuracy cannot be interpreted in isolation.

Conclusions

Problem solving for students with EBD is an emerging literature with few studies to date. This research adds to this literature base by demonstrating effects of schema instruction on the mathematic problem-solving performance for students with EBD. The study is also novel in the incorporation of problem solving, multiplication and division skills, use of effect sizes in conjunction with visual analysis, measures of social validity, and maintenance data. Schema instruction for problem solving is likely to eventually become an evidence-based practice, and additional studies toward that determination are important. Teachers in search of instructional strategies, institutions that prepare pre-service teachers, and future researchers examining the implementation of academic interventions in mathematics may consider the application of schema instruction to improve the problem-solving performance of students with disabilities.

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Corey Peltier

Kimberly J. Vannest

Texas A&M University

Address correspondence to: Corey Peltier, Department of Educational Psychology, Texas A&M University, 4225 TAMU, College Station, TX 77843-4225. Phone: (401) 487-0921. Email: coreyl024@tamu.edu.
Table 1
Sample Baseline Probe

Mr. Travis has 25 cents in his     LeBron James scored 42 points in
pocket. The cashier gives him 14   last night's basketball game. He
more cents and he places this in   scored 11 more points than Steph
his pocket. How many cents does    Curry. How many points did Steph
he have in his pocket now?         Curry score?

Ms. Lacativo has 49 M&Ms in her    There are a total of 74 students
bag that she bought. Her sister    in the cafeteria. 44 of the
had 24 M&Ms in her bag. How many   students are boys and the rest
more M&Ms does Ms. Lacativo have   are girls. How many girls are in
compared to her sister?            the cafeteria?

There are a bunch of books on      There are 22 players on the
the shelf. 16 of the books are     football team. A couple more
hardcover and 21 of the books      players join the team a week
are paperback. How many total      into the season. There are now
books are on the shelf?            35 players on the team. How many
                                   players joined the team a week
                                   into the season?

Table 2
Essential Components of Each Schema

                  Part-Part-Whole   Change            Compare

Characteristics   Static            Change over       Static
                  situation         time              situation

                  A quantity        A start           Quantities
                  comprised of      quantity          being compared
                  parts             increases or      to one another
                                    decreases

Identification    Is it static?     Is there a        Is it static?
Questions                           value changing
                                    over time?

                  Is there a whole                    Are there two
                  amount broken                       values being
                  into parts?                         compared to one

Schematic         [ILLUSTRATION     [ILLUSTRATION     [ILLUSTRATION
Diagram           OMITTED]          OMITTED]          OMITTED]

Table 3
Model of STAR and Schema Usage

S   Search the    Underlines key             Bret observed 12 squirrels
    problem       information                with black fur and 14
                                             squirrels with brown fur.
                                             How many squirrels did
                                             he observe?

T   Translate     Identify appropriate       [ILLUSTRATION OMITTED]
    the problem   schematic diagram,
                  plug in the information

A   Answer the    Identify the operational   14 + 12 = ?
    problem       sequence to compute        ? = 26 squirrels
                  the solution

R   Review the    Review the solution for    26 - 14 = 12
    problem       reasonableness             26 - 12 = 14

Table 4
Learning Objectives for Intervention Lessons

Lessons #   Topic            Learning Objective

1-2         Schema           Given a word problem, you will be able to
            Identification   match the word problem to its given
                             schema and schematic diagram.

3-5         Part-Part-       Given a part-part-whole word problem, you
            Whole            will be able to fill in the schematic
                             diagram with the given information and
                             solve for the solution.

6-8         Change           Given a change word problem, you will be
                             able to fill in the schematic diagram
                             with the given information and solve for
                             the solution.

9-11        Compare          Given a compare word problem, you will be
                             able to fill in the schematic diagram
                             with the given information and solve for
                             the solution.

12-14       Mixed            Given a word problem, you will be able to
                             choose the appropriate schematic diagram,
                             fill in the appropriate information, and
                             solve for the solution.

15-18       Equal            Given an equal groups word problem, you
            Groups           will be able to fill in the schematic
                             diagram with the given information and
                             solve for the solution.
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Author:Peltier, Corey; Vannest, Kimberly J.
Publication:Education & Treatment of Children
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Date:Nov 1, 2016
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