ABSTRACT:
This article describes a single-subject study of the effects of a contextualized instructional package on the area and perimeter performance of middle school students with emotional and behavioral disorders (EBD). The instructional package included area and perimeter problem-solving activities embedded in contextualized, or real-life, scenarios; use of manipulative techniques to reinforce concepts; and a self-monitoring component, a critical behavior management component to instructional interventions for students with EBD. Results indicated that the instructional package was effective for improving area and perimeter performance among the participants; transfer and maintenance results were mixed.
Students with emotional and behavioral disorders (EBD) experience deficits in mathematics achievement when compared with peers without disabilities (McLaughlin, Krezmien, & Zablocki, 2009; Trout, Nordness, Pierce, & Epstein, 2003; Wagner, Kutash, Duchnowski, Epstein, & Sumi, 2005). In addition, secondary students with EBD are more likely than their peers with other disabilities to receive a poor grade, be retained in a grade, and drop out of school before graduation (Wagner & Cameto, 2004). Furthermore, secondary general educators and special educators reported teaching mostly basic arithmetic skills rather than essential conceptual knowledge and skills to secondary students with EBD (Maccini & Gagnon, 2002). The literature clearly shows that the population of secondary students with EBD has limited access to and limited success with noncomputational mathematics concepts, including knowledge of area and perimeter problem solving.
Noncomputational mathematics includes complex skills and concepts related to geometry and algebra. Recent reforms in mathematics (i.e., National Council of Teachers of Mathematics [NCTM] Principles and Standards, 2000; National Mathematics Advisory Panel, 2008) and changes in general and special education regulations (Individuals with Disabilities Education Improvement Act of 2004 [IDEIA]; No Child Left Behind Act of 2001 [NCLB; 2002]) have identified the need for effective mathematics instruction of noncomputational skills, including the calculation of perimeter and area of polygons, for students with EBD in secondary grades. Area and perimeter are traditionally taught through the use of teacher-directed instruction of formulae. Often, students do not learn or understand the underlying concept associated with the formula and fail to master the formulae. This lack of mastery impairs the ability to generalize the concepts and impedes the learning of more complex geometric skills. As a consequence, students with EBD who do not acquire these foundational skills do not succeed in middle school and high school mathematics and do not meet the rigorous standards mandated by the NCLB (2002).
One of the problems with the current model of teaching geometry to students with EBD is the failure to meld the abstract concepts with the concrete applications of perimeter and area formulae. Situating learning experiences in contextualized and authentic problems can assist secondary students with EBD in understanding noncomputational concepts related to measurement and geometry (Goldman, Hasselbring, & The Cognition and Technology Group at Vanderbilt, 1997). One way to teach conceptual understanding is to give students the opportunity to construct their own knowledge, building meaning from their own background knowledge and using reallife activities to support further understanding (Goldsmith & Mark, 1 999; Maccini & Gagnon, 2002; Schloss, Smith, & Schloss, 2001; Stiff, 2001; Ward, 2001). However, these constructive approaches alone will not be adequate for students with disabilities who typically benefit from direct instruction (Baker, Gersten, & Lee, 2002; Landrum, Tankersley, & Kauffman, 2003; Maccini & Gagnon, 2000). Instead, a balanced approach that includes opportunities for students to develop deep understandings of concepts through embedded meaningful contexts associated with constructivism, while efficiently processing material through direct instruction (Smith & Geller, 2004), is necessary for teaching geometry to students with disabilities.
Currently, mathematics instruction in special education classrooms tends to focus on computation, neglecting conceptual understanding (Gersten & Chard, 1999). Consequently, outcomes are particularly poor for secondary students with EBD. Researchers found that students with EBD did not significantly improve math achievement scores despite 5 years of full-time special education services (Anderson, Kutash, & Duchnowski, 2001) and that students with EBD actually experienced an increase in mathematics deficits over their school careers (Nelson, Benner, Lane, & Smith, 2004).
Despite the known mathematic deficiencies of youth with EBD, little attention has been paid to developing and implementing effective mathematics interventions for secondary students with EBD. In a methodological review of mathematics intervention research for secondary students with EBD, Mulcahy and Krezmien (2008) found that only 1 1 studies have been conducted since 1 975. Furthermore, previous research on math interventions with secondary students with EBD focused primarily on basic skills and computation (Mulcahy & Krezmien, 2008). These earlier investigations did not examine student performance on math skills and concepts required in middle and high school settings and essential to meet national and state standards as required by NCLB (2002), IDEIA (2004), and NCTM (2000). Futhermore, there are currently no published studies on effective interventions for geometry instruction for secondary students with EBD.
The current study was designed to address these gaps in the research literature by applying a package of empirically based practices to a geometry-related topic from the middle school curriculum. Because the research literature is devoid of a set of empirically based interventions for secondary students with EBD, the intervention package was developed based on research of effective interventions for students with other disabilities (Maccini, Mulcahy, & Wilson, 2007). The study, which was conducted with secondary students with EBD, examined the effects of a contextualized instructional package on objectives related to area and perimeter. The intervention involved components of instruction that were found to be effective in previous research: (a) the use of contextualized instruction (Bottge, 1999; Goldman et al., 1997), (b) self-monitoring of student academic and social behavior (Carr & Punzo, 1 993; Reid & Harris, 1993), and (c) the use of manipulative materials and cue cards (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Cade & Gunter, 2002; Carr & Punzo, 1 993).
The study investigated four research questions:
1. Does a contextualized instructional package to teach the geometric concepts of area and perimeter result in increases in the mathematics accuracy of secondary students with EBD?
2. Do secondary students with EBD maintain performance on geometry-related tasks mastered through the use of a contextualized instructional package over time?
3. Do secondary students with EBD transfer performance on geometry-related tasks learned through the use of a contextualized instructional package to mathematics problems with similar contexts?
4. Do secondary students with EBD transfer performance on geometry-related tasks mastered through the use of a contextualized instructional package to more complex area and perimeter problems?
Method
Participants
Four middle school students with EBD (Carlos, Grace, Steven, and Riley) participated in this study. Two of the initial 4 participants dropped out of the study, and 2 additional students were recruited. The intervention with the initial 2 participants (Carlos and Grace) was replicated with Steven and Riley after Carlos and Grace completed the intervention. This was necessary to improve the power, as recommended by Tawney and Gast (1984). Information obtained from school records regarding demographic characteristics of the participants including gender, race, age, grade level, IQ, and achievement scores at the time of the study are displayed in Table 7. In addition, further description of participant behaviors and performance during the intervention are described in the Results section.
Participants met the following eligibility criteria established prior to the study: (a) They were previously school identified as having an EBD, (b) they received instruction in a selfcontained special education math class at the time of the study, and (c) they scored below 60% on an investigator-developed pretest targeting objectives related to area and perimeter of squares and rectangles.
Instructor and Setting
The intervention was implemented by the first author. The participants were pulled from a social skills class for students with EBD in a public middle school in suburban Maryland. The room used for the intervention, with two exceptions, was a 5-foot � 8-foot vacant room. The school supplied one student desk, two chairs, and two stools for the intervention. All study activities, including dependent measures, were completed in the same room. Sessions 8 and 9 for Riley were conducted at Riley's home because they fell during the school spring break. These sessions were conducted at the dining room table, with the same materials and manipulatives used at school. No other individuals were around, and the space was clear and neat during these sessions.
Measures
Domain probes, objective probes, and transfer and maintenance probes were administered to participants in the study. Domain probes provided a pretest-posttest assessment of student skills across the entire unit, much like a survey curriculum-based assessment (CBA; Hudson & Miller, 2006). Objective probes provided a daily progress monitoring of student performance on individual objectives, similar to an untimed focused CBA (Hudson & Miller, 2006). Transfer and maintenance probes assessed participants' ability to use the skills and concepts they acquire to solve problems of similar context and more complex problems, as well as the ability to continue to successfully solve area and perimeter problems over time. All probes were scored for the percentage of accurate responses.
Prior to the investigation, the domain and objective probes were piloted with a group of middle school students with average to aboveaverage academic achievement from three different middle schools in the same district. The piloted probes were evaluated for their appropriateness for use with the district's middle school population through test-retest reliability measures. Permanent products (probes) were analyzed as repeated measures of percentage of accurate responses and the percentage of items completed on the daily self-monitoring recording sheet. In addition, each student involved in the pilot was individually interviewed following the probe administration for feedback on improving the directions, nature of the questions, and physical layout of the probe.
Domain probes. The domain probes were administered to students as a pretest measure, during baseline, and after the intervention phase. The domain probes measured student performance on prerequisite tasks, such as ordering shapes by area and perimeter, as well as on tasks derived from the middle school curriculum, such as exploring the relationship among area, perimeter, and dimensions of squares and rectangles. Probe items directly related to the school district's curriculum, state curriculum, and NCTM (2000) standards. Problems on the assessment were restricted to those involving rectangles consistent with the instructional unit. The assessments consisted of (a) closed-ended questions, (b) openended questions, (c) one-step problems, and (d) multistep problems (Balanced Assessment Project Team, 1999; Hudson & Miller, 2006; Rectanus, 1997).
Parallel versions of domain probes consisted of 10 curriculum-based problems on area and perimeter, with items sampled from each objective, and totaled 42 points. A minimum of two domain probes were administered to students during baseline. Following the intervention phase, three domain probes were administered on successive school days.
Objective probes. Parallel versions of the objective probes were given only during the intervention phase, at the end of each class session in which the particular objective was taught. Objective probes totaled 10 to 12 points each. If a participant met the criterion (80%) on the objective probe at the end of the lesson, the next consecutive lesson was administered. If the criterion was not met, the same lesson was administered the following session.
Transfer and maintenance probes. Each participant was assessed on his or her ability to generalize, or transfer, the knowledge learned on similar mathematics tasks that involve different story lines than those addressed in the instructional set (near transfer) as well as more complex and different math problems than those targeted for instruction (far transfer). Participants were also assessed on their ability to maintain their new skills over time. A transfer and maintenance probe was administered twice during baseline and then three times following the end of the intervention. The first postintervention transfer and maintenance probe was administered on the school day immediately following the final intervention session. The transfer and maintenance probe was also administered 5 and 10 school days after the end of the intervention phase.
Social validity measure. Social validity measures ensure that research is considered by consumers to be useful (Lloyd & Heubusch, 1996). The participants completed an investigator-developed social validity instrument at the end of the study that involved questions about the appropriateness of the intervention and the participants' perceptions of its effectiveness. The instrument was based on instruments used in previous research (Butler et al., 2003; Calhoon & Fuchs, 2003; Maccini, 1998). Participants responded to 10 questions on a five-point Likert scale concerning the effectiveness of various aspects of the intervention. Participants indicated a score of 1 if they strongly disagreed with a statement, 2 if they disagreed, 3 if they felt neutral, 4 if they agreed, and 5 if they strongly agreed. All of the questions were positively phrased to maintain consistency. In addition, participants responded to six open-ended questions.
Independent Variable
The independent variable consisted of an instructional package designed to teach area and perimeter objectives drawn from the NCTM (2000) standards, the state curriculum, and the district curriculum. The independent variable targeted participants' procedural and conceptual knowledge of area and perimeter through contextualized problem-solving opportunities and included a self-monitoring behavior management component.
Contextualized instruction. Contextualized instruction was presented in the form of real-life applications of meaningful area and perimeter problems (Bottge, 1999; Goldman et al., 1997) provided in each lesson. For example, topics were introduced and practice opportunities were provided related to building patios, designing skate parks, building dog pens, and other real-life contexts.
Procedural knowledge. Procedural knowledge was targeted through the use of cue cards that identified the process learners use to solve problems (Cade & Gunter, 2002; Carr & Punzo, 1993; Hawkins, Brady, Hamilton, Williams, & Taylor, 1994; Hudson & Miller, 2006). Cue cards included definitions of area and perimeter and contained space for students to write in formulae and procedures during instructional sessions. Participants were taught to use the cue cards to assist them in solving problems during the intervention phase.
Conceptual knowledge. Conceptual knowledge was targeted through the use of concrete applications (manipulatives) and pictorial representations during instruction. Participants were required to solve problems using manipulative materials including Geoboards, Post-it notes, inch tiles, and paper clips. Participants identified a variety of rectangle- and squareshaped objects in the room and then measured and calculated the area and perimeter of those shapes using the manipulatives to enable students to develop a conceptual understanding of the relationships between area and perimeter of similar geometric figures. Students then used the conceptual understanding gained from practice with concrete examples to develop and understand the use of formulae at an abstract level. When formulae are understood through this process, participants should be able to solve area and perimeter problems in a more efficient manner (Hudson & Miller, 2006).
Self-monitoring. Self-monitoring of academic performance and behavior was included as a component of the instructional package. Participants were presented with a self-monitoring checklist derived from materials and procedures used in prior research on self-monitoring (Carr & Punzo, 1993; HaIIahan, Lloyd, Kosiewicz, Kauffman, & Graves, 1979; Harris, Friedlander, Saddler, Frizzelle, & Graham, 2005; Reid & Harris, 1 993; Uberti, Mastropieri, & Scruggs, 2004) and were taught to self-record the total number of problems completed and the number of problems completed correctly during independent practice for each session. The selfmonitoring checklist also required students to self-evaluate based on a set of behavioral expectations.
Instruction on self-monitoring occurred at the beginning of the intervention phase. First, each participant was provided with explicit definitions of "academic achievement" and "on-task behavior." Then, behavioral expectations were discussed, including a definition of "appropriate behavior," and examples and nonexamples were provided of desired behaviors. Each participant had an existing behavior contract; the instructor referred to the behavioral expectations in the contracts to provide a rationale for self-monitoring behavior. Each student worked on individual behavioral expectations through use of the contract. Procedures for completing the self-monitoring checklist were modeled by the instructor and then practiced by the participants.
Instructional Procedures
Components of effective instruction (Hudson & Miller, 2006; Rosenshine & Stevens, 1986) were used in instructional delivery. Lesson plans were structured to include each component to ensure a systematic implementation of the lessons. New concepts were presented in short, concrete steps, as recommended by Hudson and Miller (2006) and Rosenshine and Stevens (1986). Constructivistbased activities were infused into each lesson during demonstration and guided practice opportunities throughout the unit.
Instructional unit and lesson plans. The instructional unit, including all lesson plans and activities, was investigator developed and adapted from prerequisite and gradeappropriate concepts and skills within the NCTM (2000) standards, the state curriculum, and the district curriculum, but the instructional content was adapted to meet the academic needs of the participants. Adapting instructional content is often necessary (a) to meet the individual needs of students with EBD, (b) to teach related prerequisite skills, and (c) to access grade-appropriate mathematics standards (Bryant, Kim, Hartman, & Bryant, 2006). For instance, the unit addressed only customary units of inches and feet; the problem examples focused only on rectangular regions and involved two-digit by two-digit computation. These adaptations were implemented to ensure that participants would not experience difficulties with other mathematics skills and concepts that would interfere with their ability to learn area and perimeter.
The instructional unit consisted of 1 1 lessons, including the introduction/self-monitoring lesson. The remaining 10 lessons covered topics across four objective sets. The first 8 lessons (Lessons 1-1 through 3-2) focused on prerequisite skills, whereas the last 3 lessons (Lessons 4-1 through 4-3) dealt with middle school objectives. The length of the intervention was designed to reflect the length of a typical secondary mathematics unit. The intervention included 45-minute instructional periods, 4 days per week, over an average of 1 1 .5 sessions (r = 11-13) per participant.
Experimental Design and Study Procedures
The current research study employed a single-subject multiple probe across 2 participants, replicated by 2-participants design (Horner & Baer, 1 978; Kennedy, 2005; Tawney & Gast, 1984) to evaluate the effects of teaching secondary students a contextualized instructional package with self-monitoring techniques to solve geometry problems. The multiple-probe technique was chosen because it allowed the researcher to conduct infrequent probes of baseline behavior rather than continuous baseline measures in instances in which continuous measurement may be reactive or impractical or when an assumption of baseline stability can be made in advance of the study (Horner & Baer, 1978). In the current study, collecting continuous baseline data was both impractical and unnecessary. Because students did not receive instruction in area and perimeter during the baseline phase, if they did not demonstrate mastery of the skills at pretest, no threat could be assumed of an unstable baseline due to exposure to those or similar topics. In addition, collecting infrequent probes rather than continuous baseline data allowed data collection on a greater number of participants and was necessary because of time restrictions imposed by the school schedule.
Baseline. The baseline was established through administration of the domain probes. To ensure that regular mathematics instruction was not interrupted, the participants were administered the baseline probe sessions and intervention during a daily social skills class period. Throughout the study, participants received mathematics instruction from their special education math teacher in the selfcontained classroom on topics and activities in the district's middle school mathematics curriculum. During the baseline phase, the students had no contact with the instructor except during probe sessions.
Intervention. Each participant received the intervention after the preceding participant completed the intervention phase. Participants' out-of-class time was monitored, and sessions were scheduled so that all participants missed the same amount of the social skills class. Because instruction took place during the students' social skills class, no participants missed regular math instruction.
Booster session. An additional booster session (Montague, 2004) was added to the intervention after low performance on the first postintervention domain probe session by the first two participants. Booster sessions were the same for each participant; each objective was reviewed, and two examples were practiced from each objective set. No objective probe was administered during the booster session. Domain probes were administered in subsequent sessions.
Interrater Reliability and Fidelity of Treatment
Interrater reliability. Interrater reliability was obtained on 100% of the domain probes. A trained research assistant and the investigator independently scored each probe. The percentage of scorer agreement was calculated by dividing the number of agreements of correct responses by the number of disagreements and agreements and multiplying the result by 100% (Kennedy, 2005). Reliability was 100% across all domain probes for all students.
Fidelity of treatment. Fidelity of treatment observations were conducted by independent observers via a checklist that incorporated the components of the intervention. Prior to the investigation, the observers were trained on the components of the intervention and provided the fidelity checklist. Training consisted of an explanation of the instructional procedures, a review of the components of the fidelity checklist, and two mock instructional sessions, in which fidelity data were collected. Observers maintained 100% agreement over the mock sessions.
Fidelity observations were conducted for 33% of the instructional and probe sessions. The independent observer sat in the room with the instructor and the participant. The observer sat in a location that allowed him to see each component of the intervention, and he did not speak to the child or the instructor during the intervention. The fidelity of treatment for each session was calculated by dividing the number of components present (as recorded on the checklist) during the observation by the number of total components on the checklist and multiplying the quotient by 100%. The fidelity of treatment for the study was obtained by adding the percentages for each session and dividing the sum by the number of observations. Treatment fidelity for the study was 100%.
Interobserver agreement was obtained one time for every three observations of the first independent observer. During these sessions, a second independent observer conducted fidelity of treatment (via the checklist) at the same time as the first observer. After each observation, the observers added their completed checklists to a notebook, which was reviewed by the investigator. Interobserver agreement on the fidelity of treatment was calculated by dividing the number of agreements on the checklists from each independent observer by the total number of items on the checklist and multiplying the quotient by 100%. Interobserver agreement for the study was 1 00%.
Results
Academic Measures
Math accuracy. All participants improved their mean percentage accuracy from baseline to postintervention on the domain probes. Figure ? displays the results of percentage accuracy on domain probes during baseline and postintervention phases for the first 2 participants (i.e., Carlos and Grace). Figure 2 displays the results of percentage accuracy on domain probes during baseline and postintervention phases for participants in the replication (i.e., Steven and Riley). Visual analysis of the four graphs reveals marked improvement in level from baseline to postintervention for all participants. Furthermore, there was low variability in data points in both the baseline and postintervention phases, reflecting stable data for each participant. Carlos increased 91 percentage points in mean percentage accuracy, Grace increased 72 percentage points, Steven increased 83 percentage points, and Riley increased 67 percentage points.
An error analysis of the postintervention domain probes revealed that each participant earned point deductions for inappropriate or missing labels (e.g., labeling solution with inches or inches2). Across four postintervention domain probes, Carlos missed 3 points because of mislabeling. Across four postintervention domain probes, Grace missed 6 points because of mislabeling. Across three probes, Steven missed 2 points because of mislabeling. Finally, Riley missed 4 points because of mislabeling on three probes. Errors similar to these were also noted on the pretests; however, the more apparent issues with the pretests were the lack of completion of questions (students attempted very few items on the pretest) and errors in computation or conceptual understanding for the problems that were completed.
Each participant experienced high levels of performance across objective probe administrations. Carlos was the only participant who required repeated lesson sessions to meet 80% criterion on objective probes. He required one repeated session following Lesson 1-2: Ordering by Area. He also required one repeated session following Lesson 2-1: Measuring Perimeter Using Non-Standard Units. Both of these lessons were prerequisite skills and were administered during the 1st week of the intervention. For the remaining sessions, Carlos earned a mean score of 96.6% on objective probes (r = 90%-100%). Grace, Steven, and Riley demonstrated high levels of performance on objective probes throughout the intervention, earning mean scores of 97.5% (r = 90%1 00%), 96.2% (r = 80%-1 00%), and 98% (r = 90%-1 00%) on objective probes, respectively.
Maintenance. The maintenance task was an exercise taken directly from the instructional unit. As demonstrated in Figure 7, Grace showed no improvement on the maintenance task from baseline to postintervention sessions. In contrast, as demonstrated in Figure 2, Steven earned a score of 100% on the maintenance task during each postintervention transfer and maintenance probe session, and Riley earned a score of 0% on the first postintervention maintenance task and then 100% during the 5-day and 10-day postintervention transfer and maintenance probe sessions. Carlos refused to attempt the maintenance task during each of the postintervention transfer and maintenance probe sessions.
Transfer tasks. The near transfer task was similar in nature to tasks completed during the intervention, but participants were not explicitly taught using the exact wording or steps required to solve this particular problem. Grace and Riley each earned a mean score 0% on the near transfer task during postintervention transfer and maintenance probe sessions, and Carlos refused to attempt the near transfer task during each of the postintervention transfer and maintenance probe sessions. Steven, on the other hand, showed improvement on the near transfer task, earning scores of 100%, 83%, and 100%, respectively, on each of the postintervention transfer and maintenance probe sessions.
The far transfer task consisted of a complex, multistep problem using the same concepts and skills covered in the instructional unit. No direct instruction was provided on the far transfer task. Grace, Steven, and Riley each showed improvement on far transfer tasks, albeit limited (r = 50%-58%), from baseline to postintervention (see Figures 7 and 2). Carlos refused to attempt the far transfer task.
Self-monitoring. Each of the participants completed the self-monitoring checklist at the end of each session without instructor prompting or input. In addition, each participant appeared interested and eager to complete the self-monitoring checklist each day. Participants regularly cheered for themselves when they calculated the percentage correct.
Social validity. The average scores on the social validity measure ranged from 4 to 5 (M = 4.7, mode = 5). Overall, participants responded that they strongly agreed that the intervention package helped them to understand area and perimeter better (M = 5), and they would recommend the intervention to others (M = 4.75, r = 4-5). They also responded that they strongly agreed that they feel better about measurement and geometry skills as a result of using the intervention (M = 5). Finally, the participants reported that they strongly agreed that the self-monitoring checklist helped them to solve area and perimeter problems and they would recommend its use to other students (M = 4.75, r = 4-5).
The participants also responded positively to the intervention in the open-ended questions. For example, Riley stated she was interested in learning measurement and geometry skills "so I could help others, and teach them, too." Grace reported that "this is what I'm best at in math." Carlos stated, "If I'm needed, I will be able to help." When asked what they liked best about the intervention, participants' responses were favorable. Riley reported that she "liked using the cubes and the Geoboard," and Steven reported that he liked the cue card best. When asked what they liked least, the response was unanimous: the tests.
Discussion
This study represents the first investigation of the effectiveness of a contextualized instructional package on the area and perimeter performance of middle school students with EBD. The findings from this investigation contribute to the existing research literature base in several ways. First, prior to this study, no research had been published on mathematics interventions for secondary students with EBD in the area of geometry and measurement. Second, the instructional package studied in the investigation represents a unique combination of empirically supported components, delivered within a balanced approach integrating constructivism and explicit instruction (Mulcahy & Gagnon, 2007; National Mathematics Advisory Panel, 2008). Third, the instructional package taught prerequisite and grade-appropriate area and perimeter objectives directly aligned with the NCTM standards, the state curriculum, and the district curriculum. This indicates that the instructional approach implemented in this study was sufficient for meeting the rigorous standards associated with the NCLB but was flexible enough to support the learning of students with severe academic and behavioral deficiencies. Finally, this study is important because half of the participants were female, in contrast to the near absence of female participants in existing mathematics research in this area (Mooney, Epstein, Reid, & Nelson, 2003).
This investigation was able to demonstrate positive results regarding the first research question, which was related to mathematics accuracy on area and perimeter objectives. All participants were able to improve accuracy on both domain probes and objective probes with respect to area and perimeter, and each of the participants earned greater than 90% on at least two postintervention domain probes. The high performance during instruction and during the postintervention domain probe sessions suggests that the instructional package positively affected performance on mathematics accuracy. The consistent changes in outcomes for Carlos and Grace demonstrated that the independent variable had a positive impact on the dependent variable, which establishes a functional relationship (Kennedy, 2005). Replication of these findings with Steven and Riley demonstrated robustness of the experimental control and generality to other participants (Kennedy, 2005).
The findings with regard to maintenance and transfer probes were mixed. Three participants (Grace, Steven, and Riley) were able to maintain some level of performance over time, but for Grace and Riley, the performance did not meet the same level at postintervention as during intervention. Only Steven was able to accurately complete the maintenance task at each of the transfer and maintenance probe sessions. Grace and Riley would have shown commensurate performance on maintenance probes, but they misread the directions and accurately completed the wrong task (i.e., they solved for the largest area, not the smallest area), a mistake often made by typically developing students (Battista, 2003). Nonetheless, the 2 participants demonstrated familiarity with the relationship between area and perimeter, and both accurately completed parts of the far transfer task that involved calculating area and perimeter of a garden.
Carlos refused to complete the maintenance and transfer probes, typical of his behavior throughout the intervention and in the classroom. He refused to attempt novel and challenging tasks, characteristic of students with EBD (Gunter & Denny, 1998). His limited persistence to task and self-efficacy, typical of students with learning disabilities (Lackaye & Margalit, 2006; Meltzer et al., 2004), indicates that these factors may also be associated with poor performance on geometry tasks for students with EBD.
Finally, Steven was able to transfer successfully to a task with similar contexts, but none of the participants could successfully complete the far transfer task in its entirety. The results suggest that although participants were able to demonstrate mastery on perimeter and area objectives, their ability to maintain and transfer performance was limited. The mixed results on the transfer and maintenance measures suggest that students with EBD require explicit and sustained instruction of a variety of contexts to maintain and transfer newly learned geometry skills, a contention supported by previous research (Mooney et al., 2003; National Mathematics Advisory Panel, 2008). There are a number of possible explanations for the limited maintenance and transfer including a misplaced or inaccurate assumption that mastery will lead to maintenance and eventually transfer for students with EBD. This may not be the case for students with EBD on complex noncomputational math tasks. The fear of failure may prevent students with EBD from attempting an unfamiliar problem, even when they have the tools to complete it. This point was demonstrated most clearly by Carlos, who refused to attempt any unfamiliar problems. The other participants attempted the problems, but when a solution did not reflect what they expected, they did not attempt to find a more accurate solution.
It is also possible that the limited length of the intervention phase was insufficient to help participants improve performance on transfer and maintenance tasks. In fact, this intervention was much shorter than typical teachermediated interventions for students with EBD, which average 22 days (Pierce, Reid, & Epstein, 2004). Finally, the limited scope of the intervention, which targeted a narrow topic in geometry, did not include instruction in solving new and novel problems, nor did it teach problem-solving skills generally.
Limitations of the Current Study
The limitations of the current study include attrition, design flaws, and the possible effects of extraneous variables such as the location of the intervention and instructor bias. First, participant attrition led to a design change. The original study proposal involved a multiple-probe design across 4 participants, but 2 of the original participants left the study during the baseline phase because of emotional or behavioral crises including hospitalization. Thus, the study design was revised to the multiple-probe design across 2 participants, replicated by 2 participants. Although the original design was more desirable in terms of power and control, the study demonstrated power and control through replication across participants within the experiment and across experiments with 2 new participants (Kennedy, 2005). Two additional limitations to the design warrant consideration. First, one additional consecutive data point should have been collected with Grace during baseline. Although her baseline was consistent and level, Horner and Baer (1978) recommended collecting at least one additional consecutive, or true, baseline session from the previous participant, to be consistent with multiplebaseline design. Concerns about possible attrition led to the decision to begin the intervention prior to the collection of an additional baseline data point, consistent with recommendations of Tawney and Gast (1 984), who provided examples that did not include an additional consecutive baseline.
Next, changes to the instructional procedures during the study may have affected the findings. Booster sessions were not a component of the original design but became necessary as the first 2 participants were unable to demonstrate proficiency on their initial postintervention domain probes, despite performing at consistently high levels on objective probes throughout the intervention. Their difficulty with the initial domain probes may have been due, in part, to breaks in the schedule and the length of the intervention itself. The booster session served as a review, similar to a review session a classroom teacher would implement prior to a unit test (Montague, 2004). Both participants in the original sample (i.e., Carlos and Grace) increased their performance to 90% or greater after the implementation of the booster session. In the replication (i.e., Steven and Riley), the booster session was implemented in the first session following the end of the intervention. Both participants earned 90% or greater on all postintervention domain probes.
An error analysis of the postintervention domain probes revealed several point deductions due to lack of labeling or mislabeling responses despite having accurate conceptual understanding and computation. This finding suggests that the participants were able to accurately solve the problems but made careless mistakes that resulted in inaccurate solutions to the problems. As a consequence, the outcomes on probes may underrepresent their understanding of the concepts learned as they reflect mistakes that were not directly related to the study objectives. Participants were not explicitly taught the practice of reviewing problems and checking for labels within this intervention. Systematic instruction and explicit reinforcement of this practice are likely necessary for students with EBD. Alternately, scoring based only on accuracy of numeric responses, rather than making deductions for labels, might have been more appropriate.
The current study was conducted in an isolated setting, with a 1-to-1 student-toteacher ratio. There were limited opportunities for distractions from objects of other individuals. In addition, the students had the sole attention of the instructor, which is atypical of a classroom environment. Numerous distractions are typical in classroom settings, and individual students do not typically receive the undivided attention of a teacher. Consequently, the intervention implemented in this study may not be effective in group settings or in typical classrooms.
Recommendations for Future Research
This intervention was successful for the participants in a one-on-one isolated setting, when used with geometry and measurement concepts. Replicating the package with different math topics with other students with EBD at the middle and high school levels would be necessary for generalization. Once the effectiveness of the instructional package has been demonstrated through replications with different populations of students, with a variety of math topics, and in more typical classroom environments, future research should also focus on the effectiveness of individual components of this intervention package. Once this intervention has been established as empirically supported through replication and further research on the components, the important components should be examined using an experimental group design as recommended by Odom and colleagues (2005) and the National Mathematics Advisory Panel (2008).
Recommendations for Practice
The current study has implications for teachers of secondary students with EBD. First, results of this study indicate that students with EBD can master grade-appropriate objectives when presented through the use of an instructional package including contextualized instruction and self-monitoring of behavior and academic performance. The intervention was implemented over a period of time that is reflective of the length of a typical secondary instructional unit, and the instructional package may be effective for inclusive classrooms, special education resource rooms, and selfcontained classrooms, when a teacher is able to provide explicit and sustained instruction to individual students. However, the intervention may be impractical for classrooms in which teachers are pressured to cover a vast amount of grade-level material in a limited amount of time and may be restricted in the amount of prerequisite material that can be covered, such as general education classrooms.
Conclusion
Access to the general education curriculum and high-quality instruction in geometry is necessary for improved mathematics performance among students with EBD. Federal policies support the need to improve mathematics performance for secondary students with EBD (IDElA, 2004; NCLB, 2002; NCTM, 2000), but the field has done little to respond to the specific needs of this population. Prior to this study, research on interventions targeting measurement and geometry with secondary students with EBD was absent from the research base. This study provides seminal evidence that these students can improve performance on grade-appropriate measurement and geometry objectives when instruction is delivered through a contextualized instructional package, despite academic and behavioral difficulties. The participants in this study also demonstrated limited success in maintaining performance over time and transferring performance to novel and more challenging tasks.
Continued research is critical to identify effective approaches to teaching geometry to secondary students with EBD. A set of empirically supported instructional approaches for this population of students will likely contribute to more favorable outcomes, including improved performance in math classes, increased high school completion rates, and improved performance on national and international mathematics assessments. With increased demands for mathematics proficiency in school and in the workplace, secondary teachers of students with EBD can no longer focus efforts on basic computation skills to the exclusion of teaching students the concepts and problem-solving skills associated with noncomputational mathematics. Secondary students with EBD can master complex abstract mathematics, but innovative and comprehensive instructional approaches must be developed, studied, and implemented to achieve this end. The field laudably continues to dedicate resources toward research to prevent the development of mathematic deficiencies for primary students, but secondary students with EBD continue to struggle in mathematics and fail to meet standards and expectations in algebra and geometry. In light of what we know about long-term outcomes for students with EBD, the failure to dedicate adequate energy and resources to interventions for secondary students is no longer acceptable.
[Reference]
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[Author Affiliation]
Candace A. Mulcahy
State University of New York at Binghamton
Michael P. Krezmien
University of Massachusetts, Amherst
[Author Affiliation]
AUTHORS' NOTE
Address correspondence to Candace Mulcahy, School of Education, State University of New York at Binghamton, PO Box 6000, Binghamton, NY 13902; email: cmulcahy@binghamton.edu.
MANUSCRIPT
Initial Acceptance: 6/19/09
Final Acceptance: 7/6/09