Co-taught middle school mathematics classrooms: accommodations and enhancements for students with specific learning disabilities.

Miller, Kevin J.

Abstract

The intention of co-teaching is that the presence of two teachers with individual expertise will lead to differences in instructional delivery as compared to what is possible in a classroom taught by a single general education teacher (Friend & Reising, 1993). This study identified learning options made available to 61 students with specific learning disabilities in six co-taught middle school mathematics classrooms. Repeated observations of four co-teaching teams were conducted over a six-week period. Data collected included frequency counts for I.E.P. accommodations, instructional enhancements and flexible grouping patterns observed. A review and analysis of instructional strategies and materials revealed that, though the co-teaching model provided students with many accommodations and enhancements, implemented practices complemented traditional whole group mathematics instruction; more intensive, individualized approaches were less frequently observed.

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The National Council of Teachers of Mathematics (NCTM) (2005) envisions a future in which "all students have access to rigorous, high-quality mathematics instruction" and curriculum that is "mathematically rich, providing students with opportunities to learn important mathematical concepts and procedures with understanding." Unfortunately, emphasis on problem-solving and activity-based learning and de-emphasis on computational skills in NCTM standards has been an ongoing concern for students with specific learning disabilities (SLD), many of whom require explicit instruction and more time and practice to learn mathematics facts and computation (Miller & Mercer, 1997). Traditionally, special education research has focused on rote learning and mastery of mathematics facts and algorithms (Woodward & Montegue, 2002) based on the theory that students with SLD who lack these prerequisite skills are unable to perform higher-order mathematics (Carnine, Jitendra, & Silbert, 1997). However, mathematics instruction dominated by computational drill and practice is not consistent with the NCTM reform agenda. The lack of attention to problem solving and mathematical understanding in special education instruction raises concern that students with SLD are being denied access to the general education curriculum.

New Directions in Mathematics Instruction

Woodward and Montegue (2002) urged educators to pursue "new directions in mathematics education that will help to move students with learning disabilities out of a narrow and highly procedural set of experiences closer to the kind of mathematical instruction that is valued today" (p. 98). The need for "new directions" in mathematics instruction for students with SLD is most evident in secondary classrooms where a focus on high-level content knowledge and independent study skills is combined with a rapid pace of instruction and intensified by widespread implementation of high stakes testing (Mastropieri & Scruggs, 2001). According to the 1998-2000 Third International Mathematics and Science Study contracted by the National Center for Education Statistics (NCES), U.S. Department of Education (2003), 70% of eighth grade mathematics teachers reported curriculum guidelines and 38% reported external exams or tests played a "major role" in decisions to teach content. These findings highlight the pressing need to identify instructional approaches to support students with SLD in today's educational environment.

Evidence-based Practices in Mathematics Instruction

Researchers have identified a wide range of instructional approaches that address the needs of students with SLD in general education classrooms. The National Commission on Teaching and America's Future (2000) report, How Teaching Matters, revealed higher test scores in mathematics were associated with teacher preparation focused on individualizing instruction to accommodate the differing knowledge and skills diverse learners bring to the classroom. Tomlinson (2001) identified three elements of the curriculum that could be differentiated: (a) content (concepts, principles, and skills), (b) process (a variety of instructional groupings), and (c) products (ongoing formal and informal assessments). Multiple and flexible methods of presentation, expression, and engagement promote the achievement of individuals with wide differences in abilities (Rose & Meyer, 2002).

A number of specific approaches have been found to be successful with students with SLD. Strategy instruction, one proven approach for students with SLD (Maccini & Hughes, 2000; Montage, Warger, & Morgan, 2000; Tournaki, 2003) focuses on a series of cognitive processes with self-regulation strategies that include self-instruction, self-questioning, and self-monitoring. Studies have also documented the positive effects of manipulative instruction on the mathematics performance of secondary students with SLD (Butler, Miller, Crehan, Babbit, & Pierce, 2003; Cass, Cates, & Smith, 2003; Witzel, Mercer, and Miller, 2003). In particular, the use of manipulatives to teach students following a concrete-representational-abstract (CRA) sequence has been validated in repeated studies (Allsopp, 1999; Maccini & Hughes, 2000; Witzel et al, 2003). Allsopp, Lovin, Green and Savage-Davis (2003) identified evidence-based instructional practices in mathematics for students with SLD including: (a) teaching mathematics in authentic and meaningful contexts, (b) modeling general problem-solving strategies and specific learning strategies using multisensory techniques, (c) CRA sequence of instruction, (d) use of language to describe mathematical understandings, (e) multiple practice opportunities, and (f) performance charting. These practices emphasize learning as an interactive process in which learners construct their own understanding of subject matter as well as direct instruction for skill development, guided practice, and progress monitoring. Using a combination of teaching strategies represents a balanced approach consistent with both the NCTM reform agenda and the needs of students with SLD.

Co-teaching to Accommodate Students with SLD

Renewed enthusiasm for co-teaching models has been sparked by the most recent reauthorization of the Individuals with Disabilities Education Act, the Individuals With Disabilities Education Improvement Act (IDEIA) of 2004, passed by Congress on November 19, 2004. The final version of the bill aligned IDEIA with requirements in the No Child Left Behind Act of 2001 that by 2006, all students must have access to highly qualified teachers. In the case of the highly qualified secondary mathematics teachers, general and special educators alike must be certified in mathematics. However, one option available to the high numbers of special educators not certified in mathematics, but currently teaching secondary mathematics curriculum to students with disabilities, is to participate in collaborative teaching models (Council for Exceptional Children, 2005).

According to Cook and Friend (1995), "Co-teaching is an instructional approach in which a general educator and a special educator, share the responsibility for planning, instructing, and assessing a diverse group of students, some of whom have unique educational needs, in a single physical space" (p. 1). For over a decade, co-teaching has been touted as a solution to the dilemma in which low-performing students at the secondary level are often taught by either general education teachers who are increasingly focused on content and curricula, but have less preparation in differentiating their instruction; or special education teachers who have the skills to adapt and individualize instruction, but lack subject area knowledge (Dieker & Murawski, 2003). General educators approach their expanded roles in the planning and implementation of programs for students with disabilities with various levels of preparation, experience, and skill (Boudah et al., 2000; Dennis & Ryan, 2000; Miller, Wienke, & Savage, 2000). Minke and Bear (1996) found that teachers perceived instructional adaptations as desirable, but practical only in the context of a skilled, compatible co-teaching team. Effective teams offer high-involvement teaching strategies that afford all students in co-taught classrooms with increased opportunities for active participation and teacher interaction (Friend & Cook, 2003). The intention of co-teaching is that the presence of two teachers with individual expertise will lead to differences in instructional delivery as compared to what is possible in a classroom taught by a single general education teacher (Friend & Reising, 1993).

Purpose of the Study

This study is an outgrowth of a long-standing school-university partnership. It was designed to explore the extent to which the intention of co-teaching was realized in middle school mathematics classrooms. Specifically, the researcher sought to identify the instructional supports, accommodations and enhancements made available to students with SLD in co-taught mathematics classrooms. The following research question was addressed: What are the frequencies for various grouping patterns, accommodations, and enhancements implemented by co-teaching teams in middle school mathematics classrooms as measured by the Co-teacher Roles and Responsibilities Inventory (CRRI)? The investigation involved a review and analysis of teaching methods, materials, accommodations, and enhancements provided in six co-taught middle school mathematics classrooms.

Methods

Description of the Setting

Administrators and teachers in a Central Florida middle school expressed an interest in further developing their co-teaching model through professional development. The participating middle school earned a state letter grade of "A" based on school-wide performance on 2003 state assessments the prior year. Student enrollment at the beginning of the 2003 school year was 1,763 with approximately 55 percent Caucasian, 30 percent Hispanic, 10 percent African American, 7 percent Asian, and 1 percent in other categories. The attendance zone was considered middle to high socioeconomic status with 25 percent of the student population enrolled in the free or reduced lunch program. A continuum of services was provided to students with SLD including, (a) self-contained and pullout classes for language arts, mathematics, and science, (b) co-taught classes for mathematics and language arts, and (c) consultative services in social studies, science, language arts, and mathematics.

Description of the Participants

Four co-teaching teams, two at the sixth and two at the seventh grade level, agreed to participate in the study. Each team consisted of a special education teacher and a general education mathematics teacher. Each of the two sixth grade teams co-taught two mathematics classes; and each of the two seventh grade teams co-taught one mathematics class. Sixth grade teams followed the same daily lesson plan for both of their co-taught mathematics classes. A total of sixty-one students identified with SLD, based on Florida criteria, were enrolled in the six mathematics classes. According the Florida Department of Education (2000), "Specific learning disabilities refers to a heterogeneous group of psychological processing disorders manifested by significant difficulties in the acquisition and use of language, reading, writing, or mathematics. These disorders are intrinsic to the individual and may occur across the life span. Although specific learning disabilities may occur concomitantly with other handicapping conditions or with extrinsic influences, the disabilities are not primarily the result of those conditions or influences" (p. 171). All sixty-one students had I.E.P. goals addressing mathematics.

It was the intention of the model that approximately ten students with SLD and 20 students without disabilities would be assigned to a single class. In reality, numbers of students with SLD varied from one co-taught classroom to another (Team 1, seventh grade, one co-taught class, n=11; Team 2, seventh grade, one co-taught class, n=8; Team 3, sixth grade, two co-taught classes, n=10, n=l1; Team 4, sixth grade, two co-taught classes, n=10, n=11).

All eight teachers held undergraduate degrees in education and professional certification. Two special educators (Team 1 and Team 2) and one general education (Team 4) held Master's degrees. General educators teaching mathematics at the sixth grade level were certified in elementary education rather than in mathematics. Teaching experience ranged from 2 to 22 years. One special educator (Team 2) had 12 years of co-teaching experience. The remaining seven co-teachers had from one to four years co-teaching experience.

Co-teaching teams were newly formed for the 2003/2004 school year. Novice teachers were generally paired with veteran teachers. Co-teaching preparation for six out of eight participants consisted solely of district and building level workshops. Only two special educators (Team Three and Team Four), both with less than five years teaching experience, indicated university coursework or internship experience in the area of co-teaching. Prior to the initiation of this study, all eight teachers participated in co-teaching workshops as a part of the co-teaching professional development provided by university faculty. A total of four workshops focusing on special and general educator collaboration and co-teaching were provided over the first half of the 2003-2004 school year prior to the initiation of this study.

Observation Instrument

The Co-teacher Roles and Responsibilities Inventory (CRRI) is an observation instrument developed for this study. CRRI items are grouped in five categories: (1.) Grouping Patterns, (2.) I.E.P. Accommodations, (3.) I.E.P. Assessment Accommodations, (4.) Other Assessment Accommodations, and (5.) Enhancements. The CRRI inventory also provides space to document mathematics topics covered and instructional patterns observed (e.g., introduction, practice, or review of new content; pre test or posttest unit review, assessment). The CRRI is depicted in Table 1.

CRRI accommodations items were identified from the Individual Education Plans (I.E.P.) of the 61 students with SLD. All I.E.P.s reviewed were computer generated. Teachers had selected accommodations from a pull-down menu in the I.E.P. software program. Some accommodations were subject specific (i.e. "mathematics gridded paper may be used"), though most were generic to all content areas (i.e. "additional time for assignments"). The number of accommodations documented on individual student I.E.P.s ranged from three to eight with a mean of five. I.E.P. accommodations designed specifically for assessment situations are listed separately. See Table 2 and Table 3 for number of students with each of the specific accommodations documented on their I.E.P.s.

CRRI enhancement and grouping pattern items represent research-based best practices for students with SLD in mathematics including; (a) cooperational instructional strategies (Vaughn, Bos, & Schumm, 2003) (b) advance organizers, think alouds, independent and peer-tutoring work, and charting performance (Allsopp, 1999; Allsopp et al., 2003); (c) learning strategies (Maccini & Hughes, 2000; Montegue, 1992; Montague et al., 2000; Tournaki, 2003); (d) concrete manipulations and pictorial representations (Butler et al., 2003; Cass et al., 2003; Witzel et al., 2003); (e) CRA (concrete-representational-abstract) sequence of instruction (Allsopp, 1999; Maccini & Hughes, 2000; Witzel et al., 2003).

Participating teachers reviewed a draft of the CRRI in the second co-teaching workshop. Other assessment accommodations and enhancement items not documented on student I.E.P.s were added based on teacher input in response to open-ended survey questions asking for their perceptions of specific accommodations and enhancements provided to students with SLD in co-taught classrooms.

Observer Training

Observers (first author and two special educational doctoral students with classroom experience in middle school settings) were trained in the use of the CRRI. During two 3-hour training sessions, operational definitions for CRRI items were reviewed and each observer was given multiple practice opportunities using the CRRI while viewing previously videotaped co-teaching demonstrations. Each observer inventoried the grouping patterns, and specific accommodations and enhancements observed. The results were compared, discrepancies were discussed, and tapes were reviewed for clarification. One hundred percent interobserver agreement was reached in discussion following each observation.

Data Collection

The six-week study was conducted during the second quarter marking period of the 2003-2004 school year. Each of the four co-teaching teams was observed on a weekly basis for a total of five one-hour observations per team. A CRRI form was completed during each observation. The five observations analyzed for each team included four observation sessions during instructional lessons and one observation session during a testing situation. The first author conducted observations in all 20 of the classroom observation sessions selected for analysis. Frequency counts were recorded for CRRI items implemented by each team across five observations (four instructional and one testing situation). During eight of the twenty sessions (40%), the researcher was accompanied by one of the trained observers. Point-by-point comparisons between the researcher and trained observer's CRRI data revealed an average interobserver agreement of 95%. One hundred percent interobserver agreement was reached after discussion following each observation.

Results

Mathematics Content and Instruction

Students were exposed to a broad range of mathematics content. Topics covered included adding, subtracting, multiplying and dividing whole numbers, positive and negative integers, decimals, fractions, percents; algebraic expressions; solving equations; measuring lines, areas, volumes, perimeters and angles; plotting or reading graphs; applying formulas to solve real-life problems; probability; transformations; tessellations; lines of symmetry; congruence; and finding the mean, median, and mode.

Co-teaching teams administered five to six unit tests over a 45-day report card period. This translated into about eight to nine periods of instruction for each unit. Five to six periods were allocated to introduce new content, practice new content, and review new content introduced the previous day. One period was allocated for a review prior to the test day. A full period was allocated to the test and frequently the test was followed by another period for posttest review prior to beginning a new unit. Instructional periods generally began with homework checks and 10 to 15 minute review exercises. This was typically followed by 20 to 30 minutes of whole group instructions with the remainder of the period allocated to either individual or group practice. Instructional procedures observed included demonstration, guided practice, independent practice, and corrective feedback. Reviews included activities such as note-taking, study guides, review games, extra practice, and guided practice. Three out of the four teams conducted a post-test review during one of the instructional observations. These reviews generally involved going over problem solutions with the whole class or allowing students opportunities to ask questions about items that were marked incorrect. Homework was a major component of instruction in all co-taught mathematics classrooms.

I.E.P. Accommodations Provided

Table 2 provides a summary of accommodations listed on student I.E.P.s in order from most to least documented. The total number of students with each specific accommodation noted on their I.E.P. appears in parenthesis after each accommodation. Frequency counts for each accommodation provided over four instructional periods per team are shown in Table 2. Additional directions, cues, and prompts were available to all students on an as needed basis. Individualized supports and accommodations were rarely observed. During one observation session, the Team One (Grade 7) special educator sat with a new student in the class for an extended period of time working on content covered by the class that she had missed. The Team Three (Grade 6) special educator provided individual students with copies of notes. Accommodations for additional time to complete assignments consisted of in-class time to begin assignments and the opportunity to complete work at home. According to the school's definition of co-teaching, "The curriculum is offered at the same pace and intensity as the other classes in which the content teacher instructs." The school's mathematics department had developed a schedule for the introduction of specific topics at each grade level in order to address grade level Florida Sunshine State Standards prior to administration of the Florida Comprehensive Assessment Test (FCAT) in the spring. Keeping up with the pace of this school-wide schedule was an important consideration for these teachers. All four mathematics teachers posted two daily agendas, one for their regular classes and one for advanced classes. Co-taught classes followed the regular class agenda; curricular goals and objectives were the same for all students. Students with SLD received the same type of instruction, used the same materials, and completed the same assignments as those without SLD. The need for shortened assignments was documented on some student I.E.P.s (i.e. "work presented in short segments" (7 students) and "fewer practice items" (14 students), but these accommodations were not observed.

Assessment Accommodations Provided

A focus on maximizing student performance on assessments was evident in many of the observations conducted even though only one of the five observation sessions for each team was conducted during an assessment or testing situation. Teams were provided with mathematics practice books for the Florida Comprehensive Assessment Test (FCAT). Team Two (Grade 7) and Team Three (Grade 6) provided FCAT practice during all four instructional observation sessions. When administering a test, teachers provided an overview of the test, detailed directions, and cues for responding to specific test items. Co-teachers interacted frequently with students during the testing situation. They responded to students' questions, read test items aloud, and provided additional directions, cues, and prompts. All teams allowed students extended time to complete tests. This typically meant that students had the whole period to complete the test. Most students finished well ahead of time and were required to remain in their seats and work or read quietly while other students continued to work on the test. Teachers also prompted students who completed the test early to review their answers carefully. Team Two (Grade 7) announced that students had the option to finish the test with the special education teacher the following day. In general, extra time was available on an as needed basis and did not appear to be planned in advance. For example, tests were not broken down in sections and administered to specific students in two, rather than one, class sessions.

In addition to I.E.P. assessment accommodations, there were a number of assessment accommodations observed for seventh grade teams that were not indicated on students' I.E.P.s. Team One (Grade 7) allowed all students to use calculators as well as their textbooks and notes during the test. This team also indicated that the test format had been carefully designed. Since the focus of the test involved problem-solving using formulas to find perimeter, area, or volume, the general educator used numbers in the problems that could be easily calculated to reduce the potential for computation errors. Team Two (Grade 7) formatted their test for all students to provide space for calculations on the test rather than on a separate sheet of paper. This team used timer on the overhead projector to help students pace themselves during the testing situation. Team Two students were also provided the opportunity to retake the test to improve their grade at a later date upon request.

Table 3 shows the assessment accommodations observed for each team during the testing observation. For each team, the number of students with the assessment accommodation documented on their I.E.P.s is noted in parentheses. Assessment accommodations that were present for all teams during the testing sessions observed included "additional time for tests", "may have tests read", and "clarify test instructions".

Grouping Patterns

The data collected on the CRRI for instructional grouping patterns reflected the predominance (16 out of 16 instructional sessions) of a traditional model of whole group instruction followed by private work with one-on-one teacher assistance for all four co-teaching teams. Over 16 instructional observation sessions, small group instruction was implemented only once. This was a Shadow Teaching format used by the Team One special educator to assist a small group of students. In this co-teaching format, Teacher A delivers instruction while Teacher B monitors and directs the attention of a small number of students with special learning needs in the class. Teacher B later reteaches, concepts, adapts instruction, and provides more guided and independent practice for target students (DeBoer & Fister, 1995). In this case, the special educator sat in the back of the room with four students and assisted as they followed the general educator's review of homework problems. Three of the teams observed broke up whole group instruction by inserting brief learning partner activities (4 out of 16 sessions) and the two sixth grade teams used loosely structured collaborative group activities as a follow-up to whole group instruction (3 out of 16 instructional observation sessions). A peer tutor was present in the classroom during two of the observations of Team Four (Grade 6). The peer tutor circulated and assisted during independent work, but was not observed providing specifically planned individualized instruction. Table 4 provides a summary of the grouping patterns observed for each team.

Enhancements Provided

The numbers of instructional sessions in which various enhancements were observed are depicted in Table 5. Teams provided many enhancements identified as evidence-based best practices. In particular, teachers related mathematics to real life applications and involved students in discussing and writing about mathematics on a regular basis. Demonstration and guided practice were accompanied by teacher think alouds to model thought processes involved in problem solutions. During one observation session, the Team Four (Grade 6) general education teacher instructed students in verbal rehearsal of problem solving steps. Visual aids were also provided. Team One used the chalkboard, Teams Two and Three used overhead projectors, and Team Four used a Dry Erase Board to demonstrate problem solutions. Teachers sometimes drew pictures to illustrate solutions. Classroom walls were decorated with instructional posters highlighting mathematics vocabulary, concepts and strategies, but teachers were not observed referring to these posters. Lessons that incorporated concrete materials were observed two or less times per team. Teachers sometimes used objects to illustrate mathematics concepts, but students were rarely observed handling manipulatives. Teachers briefly introduced several learning strategies to help cue problem solving procedures or memorization of terms. However systematic strategy instruction was not observed; teaching methods did not require students to recite, practice or apply strategy steps in solving mathematics problems. Though computers were available in all classrooms, students were not engaged in computer-assisted instruction during any of the observation sessions. Students were given many opportunities to improve their grades. All teams provided opportunities for students to earn extra credit in order to boost grades. Three teams provided a project assignment or alternate performance assessment.

Discussion

The rationale for the co-teaching models observed was based on the premise that the general educator and the special educator combine individual expertise to benefit all students. According to the school's definition, "Co-teaching is a partnership between the content specialist and the learning specialist." Ideally, skilled special and general educators bring not only an extra pair of hands, but also highly specialized repertoires of instructional techniques to co-teaching relationships (Friend & Cook, 2003). The special education teacher provides additional support in the classroom and supplements the content area knowledge of the general education teacher with knowledge and expertise related to teaching students with disabilities (Rea et al., 2002). This expertise includes an understanding of how disabilities impact academic performance as well as knowledge of specific instructional practices, accommodations, and enhancements to increase access to the general education curriculum for students with disabilities (DeBoer & Fister, 1995). Ideally, the special education teacher has expertise in learning styles, learning strategies, behavior modification, diagnostic/prescriptive teaching, and accommodations, whereas the general education teacher has expertise in content area, scope and sequence of curriculum, presentation of curriculum, large group management strategies, and an objective view of academic and social development (Basso & McCoy, 1997).

The presence of two teachers in the co-taught classrooms observed clearly demonstrated the increased opportunities for teacher-student interaction afforded by the co-teaching model. CRRI data revealed that the additional teacher attention was observed across all teams and all observations. The availability of a second teacher increased opportunities for students to ask questions, the number of cues and prompts that students received; and the amount of one-on-one teacher assistance. These additional supports were also provided in assessment situations in an effort to maximize student achievement. Overall the greatest strength of the co-teaching models observed was the high level of monitoring and assistance provided to all students.

Co-teacher Roles and Responsibilities Inventory (CRRI) data revealed co-teachers incorporated many of the accommodations identified in students' I.E.P.s. However, the lack of specificity for most of the I.E.P. documented accommodations left considerable room for interpretation. It is also possible that the pull-down accommodation menu in computer generated I.E.P.s facilitated the selection of generic accommodations easily implemented in general education classrooms. For example, the most frequently documented I.E.P. accommodation was extra time to complete assignments (52 students) or tests (50 students). This was translated into practice by allowing students to complete assignments as homework, allowing an entire period for the test or by making extra time available to complete tests upon student request. These practices all require extra initiative on the part of the student. Given that general problem areas for adolescents with SLD include difficulties with materials organization and assignment completion (Deshler, Schumaker, & Lentz, 1984), these practices may not be the most effective. More effective accommodations might include shortening assignments for students with SLD to include the most critical problems, breaking work down into shorter segments, or dividing a unit test between two sessions. However, these types of accommodations for time demands require a more individualized approach.

The findings of this investigation reveal that students with SLD were supported in the co-taught classrooms observed, however, differences in instruction in comparison to other mathematics classes taught by the general educator were less evident. Enhanced learning options or more specialized approaches specifically designed for co-taught classes were infrequently observed. The school's co-teaching definition, "Curriculum is offered at same pace and intensity as the other classes," was interpreted as meaning that co-taught mathematics classes should receive the same type of instructional delivery as other grade level mathematics classes. The observed accommodations and enhancements complemented whole group instruction; they did not involve adjustments in curriculum, presentation of subject matter, or materials for students in co-taught classes. During the observed lessons, none of the three elements identified by Tomlinson (2001) that could be differentiated (content, process, products) were substantially changed for co-taught classes. To access the most challenging content of the general education curriculum, students with SLD may require instructional approaches that incorporate the "defining features of special education" (Heward, 2003, p. 187). Specifically, teachers in co-taught classrooms may need to consider incorporating instruction that is individually planned, specialized, intensive, goal-directed, research-based, and guided by student performance. Such features are not easily implemented without major changes in traditional instructional delivery in general education mathematics classrooms.

Limitations of the Study

The study was limited to eight co-teachers and six co-taught classes in one middle school in a Central Florida school district during the second quarter marking period of the 2003-2004 school year. Observations took place over a six-week period and were limited to five per team. Concerns that five observation sessions might not present a true picture of the full range of grouping patterns, accommodations, and enhancements available to students were somewhat allayed by data collected. Similarities across observations for each team suggested that five sessions did provide a representative sample. With the exception of the observation conducted during a testing situation, co-teaching teams rarely deviated from a general pattern of instruction across the four instructional observations. While interpretations of these findings need to be cautiously generalized to other middle school settings, CRRI data revealed that instructional practices in the six co-taught classrooms observed bore a number of striking similarities to the findings of the TIMSS 1999 national study in terms of content and instructional delivery in middle school mathematics classrooms. These similarities suggest that these observations reflect instructional practice in typical middle school mathematics classrooms taught by a single general education mathematics teacher.

Conclusion

Maintaining the delicate balance between the demands of state mandated standards and curricula and individual student needs is a daily challenge for general and special education teachers engaged in various co-teaching models. Hutchinson (1992) wrote:

 Given the momentum that accompanies a major initiative like the NCTM
 standards, it is critical for researchers and practitioners to open a
 dialogue between those primarily concerned with mathematics education
 and those primarily concerned with students with disabilities. At the
 intersection are classroom teachers who do not need warnings, but
 rather, the products of a thoughtful dialogue. (p. 20)

Teachers engaged in co-teaching in secondary mathematics classrooms negotiate that intersection on a daily basis. One of the participating special educators pointed out, "The pace is too fast. I feel like they almost have it and we have to move on. They just need one more day. There is all this pressure to cover everything for FCAT. It is difficult to teach anything to mastery." This comment suggests that mastery for students with SLD may require more prescriptive approaches targeting selected skills that include challenging content. However the danger of prescriptive approaches is that the curriculum for students with SLD may regress to traditional special education mathematics focusing on what Woodward and Montegue (2002) described as a "narrow and highly procedural set of experiences" (p. 98). Finding the right balance between the needs of students with SLD and the demands of the general education curriculum in co-taught classrooms is critical.

A balanced curriculum is reflected in the following annual goal selected from one of the I.E.P.s reviewed for this study: "Student will understand the effects of operations on numbers and the relationship among these operations, select appropriate operations, and compute for problem solving." Meeting such goals requires access to teachers who, like the general educators in this study, have strong content knowledge in mathematics. It also requires the emphasis on writing and discussing mathematics that was found in these classrooms. However, while students with SLD had the benefit of the additional assistance made available through the reduced teacher student ratio in co-taught classrooms, skill mastery may require a co-teaching model that provides more intensive, individualized approaches. The infrequent use of research-based best practices such as peer mediated instruction (peer tutoring and learning partners), cooperative learning, multisensory techniques (concrete objects), concrete-representational-abstract sequence of instruction, and performance charting (Allsopp et al., 2003; Cass et al., 2003; Heward, 2003; Vaughn et al., 2003; Witzel et al., 2003) in the co-taught classrooms observed is troubling to the authors. In the Executive Summary: Principles and Standards for School Mathematics, NCTM (2000) called for equity in mathematics instruction and defined that as "high expectations and strong support for all students" (p. 2). According to the Council, "Equity does not mean that every student should receive identical instruction. Rather, it demands that reasonable and appropriate accommodations be made ... to promote access and attainment for all students." Access to "high-quality mathematics instruction" and "mathematically rich" curriculum (NCTM, 2005) for students with SLD in co-taught classrooms will require "thoughtful dialogue" and future research that focuses on how to fully incorporate research-based practices for teaching mathematics to students with SLD in co-teaching models.

Acknowledgments

The authors would like to recognize the dedication and commitment of the general and special education teachers who participated in this study. This study would not have been possible without their willingness to share information and allow the researchers repeated access to their classrooms. Most commendable was their eagerness to objectively review the findings to improve practice and results for students with SLD in co-taught mathematics classrooms.

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Cynthia E. Peal and Kevin J. Miller

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Table 1. Co-Teaching Roles and Responsibilities Inventory (CRRI)

Mathematics Topics/Focus of Lesson

Grouping Patterns
* Whole Group
* Skill Group
* Cooperative Learning
* Learning Partners
* Peer Tutor
* One-on-one adult

I.E.P. Instructional Accommodations
* Additional time for assignments
* Advance organizers
* Study guides
* Review/clarify directions
* Assistance with organization
* Guides/prompts for specified tasks
* Fewer practice items
* Assistance with note-taking
* May have assignments read
* Work presented in short segments
* Recheck for comprehension
* Concrete objects
* Pictures, graphics
* Individual planner
* Small group instruction
* Math gridded paper may be used
* Provide corrective feedback verbally
* Modified assignments as needed
* Use graphic organizers
* Materials uncluttered, highlighted
* Extra support for FCAT practice

I.E.P. Assessment Accommodations
* Additional time for tests
* May test with ESE teacher
* May have tests read
* Fewer assessment items
* Clarify test instructions

Other Assessment Accommodations
* Open Book
* Calculator
* Timer
* Retake Test
* Test Formatting

Enhancements
* Discussing Math
* Post Organizer
* Mnemonics
* Think Aloud
* Learning Strategies
* Writing about Math
* Real-life Application
* CRA sequence
* Math Games
* Technology/Multi-Media
* Computer Assisted Instruction
* Charting Performance
* Extra Credit
* Alternate Performance Assessment

Table 2. I.E.P. Accommodations Observed Over Four Instructional Sessions

 # Observation Sessions in Which
 Item Was Inventoried (#
 Students For Team With
 Accommodation Documented on
I.E.P. Accommodations (# Students With IEP)
This Accommodation Documented on IEP., Team 1 Team 2 Team 3 Team 4
n=61, Ranked Most to Least) n=11 n=8 n=21 n=21

Additional time for assignments (52) 4 (6) 4 (6) 4 (21) 4 (19)
Advance organizers (34) 0 (6) 1 (6) 0 (14) 3 (8)
Study guides (34) 1 (6) 2 (6) 1 (14) 1 (8)
Review/clarify directions (27) 4 (9) 4 (6) 4 (9) 4 (3)
Assistance with organization (21) 4 (4) 4 (1) 4 (12) 4 (4)
Peer support (18) 0 (0) 2 (0) 2 (8) 3 (10)
Fewer practice items (14) 0 (1) 0 (1) 0 (8) 0 (4)
Guides/prompts for specified tasks (13) 4 (2) 4 (0) 4 (5) 4 (6)
Assistance with note-taking (10) 0 (1) 0 (0) 1 (6) 0 (3)
May have assignments read (8) 4 (0) 4 (0) 4 (5) 4 (3)
Work presented in short segments (7) 0 (2) 0 (0) 0 (1) 0 (4)
Concrete objects (7) 1 (0) 0 (0) 2 (2) 2 (5)
Pictures, graphics (7) 3 (0) 4 (0) 4 (2) 4 (5)
Recheck for comprehension (6) 4 (1) 4 (1) 4 (3) 4 (1)
Individual planner (4) 4 (0) 4 (0) 4 (2) 4 (2)
Small group instruction (3) 1 (0) 0 (0) 0 (1) 0 (2)
Math gridded paper may be used (2) 0 (0) 0 (0) 0 (1) 0 (1)
Provide corrective feedback verbally 4 (0) 4 (0) 4 (0) 4 (1)
 (1)
Modified assignments as needed (1) 0 (0) 0 (0) 0 (0) 0 (1)
Use graphic organizers (1) 0 (0) 1 (0) 0 (0) 0 (1)
Materials uncluttered, highlighted (1) 0 (0) 0 (0) 0 (0) 0 (1)
Extra support for FCAT practice (1) 0 (0) 4 (0) 4 (0) 0 (1)

Table 3. Assessment Accommodations Observed During One Testing Session
Per Team

 Items Inventoried in Observation
 Sessions Conducted For One
 Testing Opportunity Per Team ((#
Assessment Accommodations (# Students Students For Team With
With This Accommodation Documented on Accommodation Documented on IEP)
their IEP, n=61) Team 1 Team 2 Team 3 Team 4

Additional time for tests (50) 1 (6) 1 (7) 1 (19) 1 (18)
May test with ESE teacher (12) 0 (6) 1 (2) 0 (2) 0 (2)
Fewer assessment items (6) 0 (0) 0 (0) 0 (5) 0 (1)
May have tests read (5) 1 (0) 1 (0) 1 (3) 1 (2)
Clarify test instructions (4) 1 (3) 1 (1) 1 (0) 1 (0)

CRRI Assessment Accommodations Not
Documented on Student I.E.P.s Observed

Open Book or Notes For Test 1 0 0 0
Calculator For Test 1 0 0 0
Timer 0 1 0 0
Retake Test 1 0 0 0
Test Formatting/Design 1 1 0 0

Table 4. Grouping Patterns Observed Over Four Instructional Observations
Per Team (Ranked Most to Least Observed)

Grouping Patterns Team One Team Two Team Three Team Four

Whole group instruction 4 4 4 4
One-on one with adult 4 4 4 4
Learning Partners 0 2 1 1
Cooperative Learning Groups 0 0 1 2
Peer Tutor 0 0 0 2
Skill Groups 1 0 0 0

Table 5. Enhancements Observed Over Four Instructional Observations Per
Team (Ranked from Most to Least Observed)

Enhancements/Accommodations Team 1 Team 2 Team 3 Team 4

Think Aloud 4 4 4 4
Discussing Math 4 4 4 4
Extra Credit 4 4 2 2
Writing About Math 3 3 2 3
Learning Strategies 3 3 2 2
Real-life Application 2 2 3 2
CRA Sequence of Instruction 1 2 2 1
Math Games 0 0 1 2
Post Organizer 0 1 0 1
Mnemonics 1 0 0 0
Technology/Multi-Media 0 0 0 0
Computer Assisted Instruction 0 0 0 0
Charting Performance 0 0 0 0
Alternate Performance Assessment 0 1 1 1