Middle School Mathematics Teachers' Beliefs About

Learning Disabilities Research & Practice, 21(2), 98–110 C 2006 The Division for Learning Disabilities of the Council for Exceptional Children

Middle School Mathematics Teachers’ Beliefs About Inclusion of Students with Learning Disabilities Janet R. DeSimone Lehman College, City University of New York

Rene S. Parmar St. John’s University The purpose of this descriptive study was to investigate middle school general education mathematics teachers’ beliefs and self-perceived knowledge regarding teaching students with learning disabilities (LD) in inclusive classrooms. Teacher beliefs regarding administrative support and higher education teacher preparation were also examined. The Survey on Teaching Mathematics to Students With Learning Disabilities in Middle School was completed by 228 sixth-, seventh-, and eighth-grade general education mathematics inclusion teachers from 19 states. In addition, telephone interviews were conducted with a subset of 26 survey respondents. Frequency analyses were performed on the survey data, with χ 2 tests comparing teachers on demographic variables. Follow-up interview responses were summarized to elaborate on the major research questions. The findings revealed three central issues: (1) teachers had a limited understanding of the mathematics learning needs of students with LD, (2) teacher collaboration was judged to be the most beneficial and available resource by general educators teaching students with LD in inclusive mathematics classrooms, and (3) teachers did not feel that teacher education programs at the preservice level and professional development at the inservice level were adequate in preparing them for teaching students with LD in inclusive mathematics classrooms. Implications and recommendations for teacher preparation and program implementation are provided.

Mathematics has always proved to be a challenging subject, even for general education students, in the United States. When examining the performance of students with disabilities on standardized mathematics assessments, the situation becomes even bleaker. On the National Assessment of Educational Progress (NAEP) only 6 percent of the students with disabilities who participated in the mathematics component of NAEP scored at or above the proficiency level (National Center for Education Statistics, 2004). Considering that the No Child Left Behind (NCLB) Act of 2001 mandates that all students, with only a few exceptions, master the general education curriculum, participate in standardized assessments, and achieve passing levels of performance, it becomes even more imperative to study the effectiveness of inclusion programs from a variety of perspectives. Furthermore, proportionately, students with learning disabilities (LD) are the largest special education group to be included in general education classes. Forty-nine percent of students classified with specific LD spent 80 percent or more of each school day in a general education classroom. These students are not among the groups exempt from state and national standardized tests (U.S. Department of Education, 2003). Because inclusive Requests for reprints should be sent to Janet R. DeSimone, Assistant Professor, Educational Leadership, Department of Specialized Services in Education, Lehman College, City University of New York, Carman Hall, Room B01, Bronx, NY 10468. Electronic inquiries may be sent to [email protected].

practices are rapidly growing, Cochran (1998) is correct in his assessment of the current state of education—“all teachers [have] become teachers of special education students” (p. 4). Nowadays, given the expansion of inclusion, general educators, many with little or no special education training, have been assigned the responsibility of teaching students with disabilities. One of the first steps toward understanding successful instruction in inclusive mathematics classrooms is to understand general educators’ beliefs and attitudes regarding inclusion and students with LD. Such insight can help teacher educators and staff development specialists work more effectively with teachers to develop mathematics programming that meets diverse learning needs, to design better-quality teacher preparation, and to establish needed support services at sites where inclusion programs are implemented.

Research on Teacher Attitudes Toward Inclusion There is a body of theoretical literature that posits a relationship between teachers’ beliefs and knowledge, and teaching practice (Nespor, 1987). Empirical studies based on this perspective reviewed by Pajares (1992) appear to demonstrate consistently that pedagogy is indeed affected by teacher beliefs, that beliefs are developed early in an individual’s teaching career, and are not very easy to change. Some of the research studies that investigate the link between teachers’

LEARNING DISABILITIES RESEARCH

beliefs and practices regarding inclusion are summarized below. In their book on successful inclusion programs, Kochhar, West, and Taymans (2000) reported teachers’ negative beliefs and feelings as one of the three major barriers to inclusive education. Janney, Snell, Beers, and Raynes (1995) concluded that the more experience general educators had with integrating students with disabilities into the classroom, the more positive were their attitudes. The researchers attributed the general educators’ original negative perceptions to the “confusion and uncertainty” (p. 111) that arise when objectives, policies, functions, and responsibilities are altered, sometimes drastically. In their comparison of 78 teachers who worked in inclusion programs and 84 teachers who had not yet started to teach in inclusive settings, McLeskey, Waldron, So, Swanson, and Loveland (2001) found that elementary school teachers with no experience in inclusive settings demonstrated more negative attitudes regarding school readiness, adequacy of resources, academic benefits for students with disabilities, and willingness to collaborate with special education teachers than the group of inclusion teachers. A survey of 127 teachers in grades 1 through 8 by Bender, Vail, and Scott (1995) indicated that teachers who viewed mainstreaming positively were more consistent in employing effective mainstreaming strategies than those teachers with less favorable attitudes. In their summary of 28 surveys of mainly elementarylevel general educators’ perceptions of inclusion, Scruggs and Mastropieri (1996) discovered that two-thirds of general educators supported the idea of inclusion, and half of general educators believed that inclusion is indeed beneficial for students with disabilities. However, less than one-third of the general educators thought they had adequate resources, training, and time required to implement inclusive practices successfully. Smith and Smith (2000) interviewed general education elementary school teachers about the factors that contributed to or hindered their success in inclusive classrooms. Data revealed the following four factors as having the greatest effect on general educators’ perspectives on teaching students with LD in inclusive classrooms: training (undergraduate and graduate teacher preparation and in-service programs), class load (class size, severity, and range of students’ needs), support (administration, special education department, and paraprofessionals), and time (planning lessons and collaborating with special education teachers). Although the body of research summarized above has provided numerous insights into issues to address when implementing inclusion programs, many aspects remain unexplored. First, the majority of the existing research studies focus on teachers in elementary schools, where included students are placed with a single teacher for most of the school day. In such settings, there is the potential for the general education teacher to develop a strong working relationship with the special education teacher and focus on the learning needs of a few students. In contrast, the present study examines the middle school context, where included students encounter several subject-area teachers within a single day, and mathematics teachers typically do not have long-term contact with any given student or with their special education colleagues. The structure of the elementary school allows

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for more curriculum flexibility, and included students are frequently reported to be working on materials that are below their grade placement level while placed in inclusive settings. Middle schools have a strong academic focus, teachers feel a great deal of pressure to cover mandated content, and students are expected to be responsible for their own learning to a greater degree than in elementary school. Middle school teachers are more likely to be content-area specialists and to feel responsible for developing student mastery in the area. A second limitation of existing research on teachers’ attitudes toward inclusion is that many of the current studies report attitudes of teachers toward inclusion in general. The present study focuses on the critical subject area of mathematics, where there is an increasing emphasis on student attainment as measured by standardized test scores. Third, many of the existing surveys on teacher attitudes do not specifically focus on students with mild disabilities. However, the research does suggest that teachers have varying attitudes toward inclusion, depending on the nature of the disability of the included student. Finally, the preservice and in-service experiences of elementary and middle school teachers vary greatly. While issues such as literacy development and classroom management dominate the preservice preparation programs for elementary teachers, specific pedagogy in the content area is the focus for middle school teachers. Frequently, the latter group does not feel responsible for differentiating instruction to meet diverse learning needs. The in-service programs and administrative support in middle school tend to be directed toward enhancing specific content-area instruction, not providing for diverse programming. To date, no research has specifically considered teachers’ beliefs, attitudes, and self-perceived knowledge when actually working with students with LD included in middle school mathematics classes. Purpose of the Study This descriptive study attempts to extend the existing research on teachers’ beliefs and perceptions, specifically addressing the critical area of mathematics instruction. The purpose of this study was to examine middle school general education mathematics teachers’ beliefs and self-perceived knowledge regarding teaching students with LD in inclusive classrooms, and to gain an understanding of teacher perspectives on the application of inclusion in their own schools. The study investigated the following four questions: 1. What are the general beliefs of general education middle school mathematics teachers about inclusion of students with LD? 2. What is the self-perceived knowledge base of general education middle school mathematics teachers regarding the specific learning needs of students with LD who are included in their mathematics classrooms and their ability to adapt instruction for these students? 3. What are the perceptions of middle school general education mathematics teachers regarding administrative support and resources for teaching inclusion?

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DESIMONE AND PARMAR: MIDDLE SCHOOL MATHEMATICS TEACHERS’ BELIEFS

4. What are the perceptions of middle school general education mathematics teachers regarding the preparation they have received in preservice programs to teach in inclusive classrooms? The present study reports data from a nationwide survey of middle school mathematics teachers currently teaching in inclusive settings, with follow-up information from telephone interviews conducted with a subgroup of respondents. METHOD Survey Component Surveys were mailed to 361 middle school mathematics inclusion teachers nationwide, whose names were obtained through contact with professional organizations and school districts. A total of 228 responses were received (63 percent return rate). The demographic characteristics of the respondents are presented in Table 1. The survey respondents represented 19 different states from all geographic regions of the United States. The regional breakdown was as follows: Mid Atlantic (60.9 percent), New England (19.3 percent), West (7 percent), Southwest (6.6 percent), South (4.4 percent), and Midwest (1.7 percent). Approximately 49 percent of the teachers (n = 110) were from suburban school districts, followed by 25 percent urban (n = 57) and 14.5 percent rural (n = 33). Approximately 11 percent of the respondents (n = 26) did not classify their school district. The majority of teachers taught in schools that had more than 500 students (77.7 percent, n = 177), with the average class size falling between 21 and 30 students. Approximately one-half of the respondents identified themselves as public school teachers (51.4 percent, n = 117); 1 percent (n = 2) indicated they were private school teachers; and the remaining teachers did not describe this aspect of their schools. The sample was a reasonable representation of middle schools across the country in terms of size and community location, as described by the National Center For Educational Statistics (NCES, 2003), in their report on Public Elementary and Secondary Schools. According to the NCES data, the average size for middle schools is 612 students, with 57 percent of schools being located in suburban areas, and 18 percent in major urban areas. However, it is noted that the sample was primarily from the eastern regions in the United States.

TABLE 1 Demographic Variables for Survey Respondents Variable

Number (%) a

Gender Female Male

161 (70.6) 60 (26.3)

Educational level Bachelor degree Master’s degree (completed or pursuing) Professional diploma (completed or pursuing) Doctoral degree (completed or pursuing)

24 (10.5) 182 (79.8) 16 (7) 6 (2.6)

Years of experience teaching 1–2 3–8 9–14 15 or >

41 (18) 66 (28.9) 49 (21.5) 72 (31.6)

Years of experience teaching in inclusion settings 1–2 3–5 6–10 11 or >

69 (30.3) 67 (29.4) 41 (18) 50 (21.9)

No. of math methods courses 1 2 3 4 5 6 or >

51 (22.4) 56 (24.6) 26 (11.4) 19 (8.3) 9 (3.9) 24 (10.5)

No. of inclusion- or LD-related workshops 0–2 3–4 5–6 7–9 10 or >

98 (43) 62 (27.2) 22 (9.6) 10 (4.4) 34 (14.9)

Level of support services Extremely low Low Average High Extremely high

10 (4.4) 39 (17.1) 85 (37.3) 64 (28.1) 29 (12.7)

Level of administrative support Extremely low Low Average High Extremely high

23 (10.1) 47 (20.6) 85 (37.3) 49 (21.5) 22 (9.6)

a The

number of respondents varied because of missing cases.

Survey Instrument The Survey on Teaching Mathematics to Students With Learning Disabilities in Middle School (DeSimone & Parmar, 2004) was designed as a three-part questionnaire. Part I (12 items) provided descriptive data regarding the participants and their schools, as well as participants’ perceptions of the level of administrative support and available resources for inclusive teaching (extremely low to extremely high). Part II (14 items) used a 5-point (strongly agree, agree, undecided, disagree, strongly disagree) Likert scale to measure participants’ beliefs regarding inclusive mathematics classes, stu-

dents with LD, and their prior preparation to teach in inclusive classrooms. The items from Parts I and II were adapted from existing research on teachers’ beliefs and inclusion (Larrivee & Cook, 1979; Coates, 1989; Chow & Winzer, 1992; McLeskey et al., 2001). Part III of the questionnaire had two dimensions and used a 4-point (very comfortable, quite comfortable, somewhat comfortable, not comfortable) Likert scale to assess participants’ level of comfort in their abilities to both (a) adapt their mathematics instruction for students with various LD learning characteristics (11 items) and (b) adapt their instruction


for students with LD in specific topics within the middle school mathematics curriculum (17 items). At the end of the survey, respondents were asked to list their name and telephone number if they were willing to volunteer for a phone interview. The Survey was constructed after an extensive review of literature on teacher beliefs regarding inclusion and mathematics instruction, a compilation of characteristics of students with LD found in major textbooks, and a compilation of mathematics topics from New York State (NYS) curriculum guidelines for grades 7 and 8, which were found to be largely similar across states. The NYS curriculum guidelines are consistent with NCLB goals and also incorporate research from within the United States and abroad on appropriate mathematical content (New York State Education Department, 2005, p. 2). Two procedures were used to establish the validity of the survey instrument. First, a panel of three leading researchers who had experience with teaching mathematics to students with and without LD was asked to review the survey and provide comments, resulting in some changes in wording. Second, a pilot study was conducted by administering the survey to 27 teachers in middle schools in the local area. Separate reliability analyses were conducted for the three parts, and two items were dropped to improve internal consistency, resulting in the following coefficients: general beliefs (Cronbach’s α = .75); the adaptation of instruction to fit the learning characteristics of students with LD (Cronbach’s α = .92); the adaptation of instruction to teach middle school mathematics topics effectively to students with LD (Cronbach’s α = .90); the total instrument (Cronbach’s α = .90), which were deemed acceptable for the research objectives. Follow-up Interview Component From the 42 survey respondents who had volunteered for follow-up interviews, a purposive sample of 26 was chosen from the nine states with the largest percentage of surveys. The 26 respondents, all public school teachers, were from New York (11), Rhode Island (3), New Hampshire (3), Colorado (3), Texas (2), Massachusetts (1), Pennsylvania (1), West Virginia (1), and Washington (1). The demographic characteristics of the interview participants are presented in Table 2. For the most part, the interview sample was representative of the original 228 survey respondents, though the former had more years of teaching experience and fewer years of education on average. Interview Schedule The interview schedule (see Appendix) consisted of eight open-ended (semi-structured) questions designed to provide further insight into the main constructs found in the survey. The interview schedule was initially piloted with three middle school mathematics teachers to assess the level of clarity of the questions. Interview data were collected during a span of 6 months. Participants were interviewed for approximately 30 min-

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TABLE 2 Demographic Variables for Interview Participants Variable

Number (%)

Gender Female Male Educational level Bachelor degree Master’s degree (completed or pursuing) Years of experience teaching 1–2 3–8 9–14 15 or > Years of experience teaching inclusion 1–2 3–5 6–10 11 or >

21 (80.8) 5 (19.2) 8 (30.8) 18 (69.2) 2 (7.7) 11 (42.3) 7 (26.9) 6 (23.1) 6 (23.1) 6 (23.1) 9 (34.6) 5 (19.2)

utes. Telephone interviews were not audiotaped; instead, during the interview, notes were simultaneously entered into a laptop computer. The narrative data were then analyzed using the constant comparative method (as discussed in Bogdan & Biklen, 1998), based on the major areas of the survey. Triangulation of data across sources was achieved, and peer-debriefing methods were used to establish credibility and trustworthiness of the results. The telephone interview responses clarified and elaborated upon the survey responses. Insights into challenges and success factors were obtained that could inform teacher preparation programs and classroom practice at sites where inclusive mathematics instruction is being implemented.

FINDINGS The responses received from the survey and follow-up interviews are summarized below, organized according to major sections of the survey instrument. The responses revealed several issues that would need to be addressed in order to assist middle school mathematics teachers in providing effective instruction in inclusive classrooms.

Research Question 1: Teachers’ General Beliefs About Inclusion Teachers’ reported beliefs concerning characteristics of students with LD, inclusion, and teachers’ roles and responsibilities in inclusive classrooms are presented in Table 3.

Beliefs About Including Students with LD Approximately four out of five (80.3 percent) of the survey respondents agreed or strongly agreed with the statement

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TABLE 3 General Educators’ Beliefs Regarding Inclusion and Students With Learning Disabilities Number (%)a Beliefs Statement Beliefs About Including SLD SLD should be afforded every opportunity to learn math with general ed students SLD are best taught math in inclusive classrooms Beliefs About Implementing Inclusion For the most part, middle schools are effectively implementing inclusive programs SLD will have a better chance in society learning math in inclusive classrooms than resource rooms Resource rooms are effective in meeting the math learning needs of SLD

Strongly Agree

Agree

67 (29.4)

116 (50.9)

24 (10.5)

Disagree

Strongly Disagree

25 (11)

17 (7.5)

3 (1.3)

71 (31.1)

85 (37.3)

39 (17.1)

9 (3.9)

12 (5.3)

54 (23.7)

70 (30.7)

62 (27.2)

29 (12.7)

30 (13.2)

70 (30.7)

69 (30.3)

52 (22.8)

6 (2.6)

19 (8.3)

64 (28.1)

70 (30.7)

58 (25.4)

14 (6.1)

26 (11.4)

42 (18.4)

10 (4.4)

26 (11.4)

41 (18)

4 (1.8)

29 (12.7)

103 (45.2)

49 (21.5)

Beliefs About Roles and Responsibilities of the General Educator General ed teachers are responsible 61 (26.8) 89 (39) for modifying instruction for SLD General ed teachers are responsible 63 (27.6) 94 (41.2) for ensuring that SLD succeed academically SLD cause the most behavioral 8 (3.5) 39 (17.1) problems

Undecided

Note. SLD = Students with learning disabilities. a The number of respondents varied because of missing cases.

that students with LD should be afforded every opportunity to learn mathematics with general education students. However, fewer than one-half of the respondents (41.6 percent) believed that students with LD are best taught mathematics in an inclusive classroom, and a large percentage (37.3 percent) of the respondents were still undecided on this issue. The responses indicate a conflict between beliefs regarding equal opportunity for students with LD and reservations about how this equality could be achieved when making instructional or placement decisions. Because the respondents were currently teaching in inclusive classrooms, their responses would indicate that many of them did not personally find the instructional placement to be best for the students with LD in their classrooms. Chi-square tests of significance indicate that teachers in schools with higher levels of administrative support and availability of ancillary support services were significantly more supportive of inclusion (χ 2 = 37.72, p < .001).

resource rooms in comparison with inclusive classrooms, and fewer than half (43.9 percent) of the respondents agreed or strongly agreed with the following statement: “Students with LD who are taught mathematics in inclusive classrooms will have a better chance of succeeding in society than students taught in resource room settings.” When asked to rate whether resource rooms were more effective in meeting the mathematics learning needs of students with LD, once again, results were almost evenly split; 31.5 percent disagreed or strongly disagreed; 36.4 percent agreed or strongly agreed; and 30.7 percent remained undecided. The varied responses indicated that many middle school mathematics teachers were doubtful that the resource room model effectively ensured learning of mathematics; however, they observed that students were not learning very effectively in inclusive placements either. Chi-square tests of significance once again indicated that teachers in schools with more support felt that inclusion was more effective than those in less supportive schools (χ 2 = 85.09, p < .001).

Beliefs About Implementing Inclusion Twenty-nine percent of the mathematics teachers agreed or strongly agreed that middle schools were successfully executing inclusive practices. A fairly large percentage (30.3 percent) of respondents were undecided about the benefits of

Beliefs About Roles and Responsibilities of the General Educator Approximately two-thirds of the survey respondents believed that, as general educators teaching students with LD in


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In the second section of the survey, teachers were asked whether or not they felt comfortable adapting instruction to meet specific learning needs. Table 4 presents the percentage of middle school mathematics teachers in each category of response.

cation,” and “using pictures and diagrams” were areas where teachers felt most comfortable. In all areas, responses were generally in the mid-range of somewhat comfortable and quite comfortable. Chi-square tests of significance indicated that the teachers with high levels of support in their schools felt most comfortable adapting instruction to meet student needs for all of the specific disability characteristics (χ 2 ranges from 23.97 to 43.05, p ≤ .01). Teachers with more years of experience also felt more comfortable than teachers with fewer years experience. Workshops were found to be beneficial in helping teachers work with students with LD with attention, memory, and communication difficulties. Education level and coursework in mathematics methods had minimal or no effect in this area. In the interviews, many teachers shared that they struggled with finding ways to increase their included students’ levels of motivation and attention, and they named learning characteristics as the characteristics most difficult to address. A seventh-grade teacher from Colorado said, “I guess the lack of motivation for kids [is the hardest to deal with] . . . it’s hard to jump start them.” Others indicated that “[Getting students with LD interested] takes most of my time and planning” (teacher from New York) and “It takes such an amount of energy not to let [students with LD] be apathetic” (eighth-grade teacher from Colorado).

Mathematical Learning Needs of Students with LD

Research Question 2, Part 2: Teaching Specific Mathematics Topics

inclusive mathematics classrooms, they were the ones who were primarily responsible for modifying instruction and ensuring that their students with LD succeeded academically. A large majority (66.7 percent) of respondents disagreed or strongly disagreed with the statement that students with LD cause the most behavioral problems in inclusive classrooms, though teachers with less experience were more likely to attribute behavioral problems to students with LD. A sixth-grade teacher from New York, who was interviewed, believed that she was the primary person responsible for her included students, as well as their grades. She stated, “I felt very strongly that these were my kids, and they belonged to my class.” Another teacher from New York explained that he planned all of the lessons, which included his modifications, and then the special education teacher would give him some additional ideas on how to modify instruction.

Research Question 2, Part 1: Knowledge of LD

More than half of the survey respondents described themselves as either quite comfortable or very comfortable in their abilities to adapt their instruction to meet the special mathematical needs of students with LD. However, between 5 percent and 13 percent indicated they were not comfortable in many areas. The areas of “keeping place,” “identifying symbols,” and “maintaining attention,” had the lowest ratings, followed by “memory of information” and “written communication.” Adapting instruction to help students understand “number line,” “follow a sequence of steps,” “oral communi-

Survey respondents were asked to describe their level of comfort in adapting instruction for students with LD in relation to 17 mathematical topics drawn from the NYS Curriculum guidelines (see Table 5). Specific Mathematics Topics According to survey results, the majority of general educators seemed to be most comfortable when teaching students

TABLE 4 Level of Comfort Adapting Instruction to Meet the Needs of Students With Learning Disabilities Number (%)a Learning Needs Attending to tasks Maintaining attention Keeping place on pages Identifying symbols or numerals Using a number line Recalling math facts Following a sequence of steps to solution Memory of information in word problems Oral communication Written communication Interpreting pictures and diagrams a The

Very Comfortable

Quite Comfortable

Somewhat Comfortable

Not Comfortable

41 (18) 41 (18) 43 (18.9) 36 (15.8) 59 (25.9) 46 (20.2) 49 (21.5) 36 (15.8) 53 (23.2) 43 (18.9) 50 (21.9)

88 (38.6) 85 (37.3) 82 (36) 85 (37.3) 96 (42.1) 86 (37.7) 94 (41.2) 79 (34.6) 88 (38.6) 87 (38.2) 96 (42.1)

76 (33.3) 72 (31.6) 69 (30.3) 74 (32.5) 59 (25.9) 76 (33.3) 65 (28.5) 89 (39) 74 (32.5) 74 (32.5) 64 (28.1)

16 (7) 25 (11) 29 (12.7) 29 (12.7) 10 (4.4) 16 (7) 16 (7) 20 (8.8) 10 (4.4) 20 (8.8) 15 (6.6)


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TABLE 5 Level of Comfort Adapting Instruction for Specific Mathematics Topics Number (%)a Topics

Very Comfortable

Quite Comfortable

Somewhat Comfortable

63 (27.6) 66 (28.9) 70 (30.7) 64 (28.1) 44 (19.3) 37 (16.2) 74 (32.5) 89 (39) 65 (28.5) 51 (22.4) 56 (24.6) 34 (14.9) 33 (14.5) 58 (25.4) 70 (30.7) 70 (30.7) 44 (19.3)

86 (37.7) 85 (37.3) 87 (38.2) 87 (38.2) 73 (32) 90 (39.5) 106 (46.5) 89 (39) 85 (37.3) 84 (36.8) 84 (36.8) 49 (21.5) 57 (25) 86 (37.7) 86 (37.7) 86 (37.7) 71 (31.1)

64 (28.1) 57 (25) 55 (24.1) 60 (26.3) 84 (36.8) 68 (29.8) 36 (15.8) 37 (16.2) 58 (25.4) 71 (31.1) 70 (30.7) 62 (27.2) 69 (30.3) 62 (27.2) 55 (24.1) 51 (22.4) 76 (33.3)

Reading/writing integers, rational, irrational numbers Equivalence of fractions, decimals, percents Arithmetic operations—decimals, fractions One- and two-step word problems Inverse relationships between × and /, roots, exponents Scale drawings Coordinate planes Line and bar graphs Compasses, rulers, protractors Square and cubic units Size, quantity, capacity Graphing calculators Computer spreadsheets Estimation as problem solving Identifying, describing and creating patterns One- and two-step equations Describing functional relationships a The

Not Comfortable 12 (5.3) 17 (7.5) 12 (5.3) 14 (6.1) 23 (10.1) 29 (12.7) 9 (3.9) 8 (3.5) 16 (7) 18 (7.9) 14 (6.1) 69 (30.3) 58 (25.4) 17 (7.5) 11 (4.8) 16 (7) 30 (13.2)


with LD topics where there was extensive use of visual and manipulative materials, such as locating points on a coordinate plane and interpreting line and bar graphs. They seemed to be less comfortable when teaching students with LD to use technology tools such as graphing calculators and computer spreadsheets. The following topics were rated as very comfortable or quite comfortable: describing equivalence of fractions, decimals and percents (66.2 percent), performing arithmetic operations on decimals and fractions (68.9 percent), and solving one- and two-step arithmetic word problems (66.3 percent). Yet, close to one-half (46.5 percent) of the general education teachers surveyed (n = 106) described themselves as only somewhat comfortable (33.3 percent) or not comfortable (13.2 percent) in their abilities to modify instruction when describing functional relationships to students with LD. Finally, at least one-fourth of the respondents described themselves as only somewhat comfortable in adapting their instruction for students with LD in 12 out of the 17 mathematics topics listed on the survey. Chi-square tests of significance once again indicated that the most significant factor in teachers’ level of comfort is the presence of administrative support and support services (χ 2 ranges from 22.36 to 43.78, p ≤ .01), followed by years of experience (χ 2 = 18.62 to 27.31, p ≤ .05). Specific coursework in mathematics methods and workshops did not appear to impact teachers’ ratings in this area. Interview respondents named colored markers, overhead projectors, repetitive practice, reducing the number of examples on tests and homework, small-group work, tiering lessons and spiraling homework, delivering “small pieces [and] chunking material rather than big groups” of information, and teaching “one concept at a time” (teacher from Rhode Island) as different types of instructional modifications. In addition, many teachers emphasized the use of in-

structional methods based on multiple modalities and being “very structured . . . maintain routine for these kids’ (teacher from New Hampshire). Quite a few teachers used the term “differentiated instruction” (e.g., teachers from New York and New Hampshire), yet they could not name specific strategies when asked to elaborate more on this term or provide concrete examples of differentiated instruction. Three teachers said they really did not modify instruction specifically for their students with LD, and more than half of the teachers said they used the same instructional modifications for their lower-end students (teachers from New York and Colorado). Approximately 18 interview respondents said they had not really altered the curriculum for their included students. A seventh-/eighth-grade teacher from New Hampshire said, “With all the testing we’re supposed to teach the same content and try to make the kids learn it . . . . In the middle schools, there’s a big push to get kids ready for high school.” An eighth-grade teacher from Massachusetts agreed; she stated, “[Teachers] can’t change the curriculum. Students have tests to take.” A sixth-grade teacher from Texas said, “I do the same curriculum—it just takes longer, and I teach it differently. When I run short on time, I just make the time up in another way.” A few of the teachers from Colorado said they “water down the curriculum” and try to emphasize the parts of the curriculum that have “real-world” applications (e.g., time, money). Some teachers from New York said they cover all parts of the curriculum, but do not hold the included students “responsible” or “accountable” for difficult topics such as Pascal’s Triangle, graphing calculators, and algebra. However, a few of the teachers did identify specific curriculum adaptations for their students with LD. A sixth-grade teacher from New Hampshire stated, “For fractions, I only make them do a common denominator . . . and for decimals, I only take my special ed students to the hundredth place. The curriculum requires that they go to the place of one


thousand.” A seventh-grade teacher from Colorado replied, “I stay away from dividing three-digit numbers into fivedigit ones,” while an eighth-grade teacher from Rhode Island stated that she “may not cover . . . higher-end word problems” with her inclusion classes. Alternate curricular modifications such as better integration of topics so that all topics could be covered in the allocated time, prioritizing topics to stress important concepts early in the school year, or division of topics between general education teachers and resource room or other teachers, were not mentioned by any of the respondents. Further, 10 teachers named instructional modifications again, when asked to give specific examples of curriculum adaptations. Participants from New Hampshire, New York, Texas, and Rhode Island all stressed that fractions were very difficult for students with LD to comprehend. A teacher from New Hampshire said, “Cognitively they’re [students with LD] not mature enough to understand this.” Other topics identified included word problems, decimals, equations with variables and inequalities, geometric formulas where “understanding dimensions [was] tough” (teacher from Pennsylvania), probability, and basic skills such as addition, subtraction, multiplication, and division. Research Question 3: Support Mechanisms and Resources Survey respondents and interview participants were asked to comment on the level of administrative support and resources available to aid them in teaching students with LD (see Table 6). Administrative Support Thirty-one percent of survey respondents considered the support level of their school’s administration to be low or TABLE 6 General Educators’ Beliefs Regarding Administrative Support and Available Resources Number (%)a Beliefs Statement Administrative Support In inclusive math classrooms SLD require more time from teachers than general ed students General ed teachers are given sufficient time to prepare for teaching math inclusion Resources Are comfortable team teaching math with special ed teachers a

Strongly Agree

Agree

71 (31.1) 94 (41.2)

9 (3.9)

44 (19.3)

24 (10.5) 82 (36)

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extremely low (see Table 1). Almost three-quarters (72.3 percent) of the general educators surveyed believed that students with LD required more time from teachers than general education students. However, more than half (57.5 percent) of the respondents felt that administrators did not give them sufficient time to prepare for their mathematics inclusion classes. A few of the interview participants believed that their building administration also faced monetary and resource challenges. In some situations, as one New York teacher said, “their [administration] hands are tied,” and often administrators have no choice but to implement districtwide policies. However, many participants perceived their administration as being ineffective in offering assistance and solutions. A sixth-grade teacher from New York stated, “Administration doesn’t understand the challenges of teaching inclusion.” A seventh-grade teacher from New York said, “administration was not very helpful . . . they recognize that inclusion presents problems, but they’re not quick to offer advice.” In addition, a sixth-grade teacher from New Hampshire replied, “They [administration] don’t always focus on making the teacher’s job easier . . . like freeing us up from mundane things . . . free us up for what’s important . . . they lose sight of this.” Most teachers were just told that they would be teaching mathematics inclusion and were never asked if they were comfortable with the new assignment. An eighth-grade teacher from New York criticized her administration; she said, “Teachers were not given choices . . . [some] were just assigned inclusion . . . some didn’t have any idea what this was . . . they don’t want to be teaching inclusion and didn’t have a choice.” Some of the teachers did not think the administration afforded them adequate and consistent professional development opportunities focused on inclusion. A seventhgrade teacher from Texas commented, “Administration tells you that you need to modify and [they] want to see inclusion work, but they don’t help you much. [There is] not much staff development, and it’s really sad—especially for the firstyear teachers . . . maybe 10 minutes of staff development is on special ed.” An eighth-grade teacher from Rhode Island said that her administration was “more supportive the first year [of the inclusion program] . . . no recent workshops [had been planned]” on specific instructional strategies. Resources

Strongly Undecided Disagree Disagree 24 (10.5)

33 (14.5)

43 (18.9)

83 (36.4) 48 (21.1)

76 (33.3)

37 (16.2)

The number of respondents varied because of missing cases.

5 (2.2)

8 (3.5)

When asked to rate the level of available support services (e.g., counseling, resource room or teacher, instructional materials, etc.), about one-fifth (21.5 percent) of the survey respondents felt that existing services were below average (see Table 1). Approximately 43 percent of respondents had taken fewer than three workshops related to teaching students with LD. Some of the respondents were not required to take any workshops. (See Table 1 for full details on the comprehensive list of categories and percentages.) Teamwork and collaboration seemed to be an integral resource. According to the survey, approximately 46.5 percent of the general educators agreed or strongly agreed with the following statement: “General education teachers are comfortable team teaching mathematics with special education teachers.” However, more than one fourth (33.3 percent) of

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the general educators were still undecided concerning their comfort with team teaching. Although eight of the interview participants did not have a special education teacher or aide in all (or any) of their inclusion classes, most of the interview participants identified other people (e.g., special education teachers, aides, other inclusion teachers, counselors, etc.) as the most significant resource available to them. Much of the interview data supported the theme of teacher collaboration as an important factor in successful inclusive classrooms. A sixthgrade teacher from New York commented, “[We] were given no direction for inclusion . . . I was very lucky to have three other committed educators . . . without them, I wouldn’t have gotten half the work done . . . . They were my support . . . the ones I bounced ideas off of . . . [We] shared everything.” Another sixth-grade teacher from New Hampshire stated, “It’s impossible to do inclusion without collaboration . . . they’re [team teachers] my eyes and ears . . . for what I don’t always pick up, they catch.” A New York eighth-grade teacher explained, “If you don’t have a good relationship with your co-teacher, it [inclusion] just doesn’t work.” An eighth-grade teacher from Washington said, “When there’s a strong sense of team teaching, everyone enjoys their job more, and the kids are more successful.” Other teachers relied on their peers (both special and general education teachers) for “feedback,” for the “challenges/benefits . . . of using certain lessons,” to “understand [IEP] goals and focus more for the kids,” or just to “hear other perspectives . . . sometimes you get in a fixed pattern on how you teach” (New York teachers). Finally, a sixth-grade teacher from New York summed up teacher collaboration by stating, “Unique ideas are generated through collaboration . . . [You] wouldn’t find them in any methods books.” Although some instructional resources were also discussed, such as Web sites, computers and software, overhead projectors, graphing calculators, Mimeo technology, manipulatives and other hands-on materials, the interview participants did not seem to rely on these materials as much as assistance from colleagues. Research Question 4: Strategies From Higher Education Teacher Preparation Programs Only about one-fourth (27.6 percent) of the respondents agreed that their teacher education programs helped them develop instructional philosophies related to teaching mathematics to students with LD (see Table 7). Half of the respondents thought that their teacher education programs had failed to offer specific information about the characteristics and needs of students with LD in mathematics learning or to offer specific instructional strategies for teaching mathematics to students with LD. Further, as Table 1 shows, more than half (57.1 percent) of the respondents had taken fewer than three mathematics general education methods classes. Ten percent of inclusion mathematics teachers were not exposed to any mathematics methods courses, possibly because this was not a requirement at the time they received certification. Teachers are not provided with opportunities to learn about specific characteristics and needs of students with LD. Further, they have no information on how to tailor instruction to

TABLE 7 General Educators’ Beliefs Regarding Their Teacher Preparation Programs Number (%) a Beliefs Statement Teacher ed programs help general ed. teachers develop instructional philosophies for teaching math to SLD Teacher ed programs offer specific information about characteristics/ needs of SLD in math learning Teacher ed programs offer specific instructional strategies for teaching math to SLD a

Strongly Agree

Agree

Strongly Undecided Disagree Disagree

5 (2.2)

58 (25.4)

56 (24.6)

77 (33.8) 27 (11.8)

8 (3.5)

52 (22.8)

52 (22.8)

81 (35.5) 33 (14.5)

5 (2.2)

49 (21.5)

50 (21.9)

80 (35.1) 42 (18.4)

The number of respondents varied because of missing cases.

address the specific disabilities demonstrated by students in their classrooms, particularly when covering the curriculum for upper grades. Tests of significance did indicate that teachers in more supportive schools generally rated their level of teacher preparation differently. Twenty of the 26 interviewees believed that their undergraduate and graduate schools did not effectively prepare them to teach mathematics inclusion. Approximately seven of the participants said that they learned “a little” or rated their teacher preparation classes as “fair.” One eighth-grade teacher from New York summed up the issue by saying, “My programs prepared me sort of . . . not great but not terrible.” Another eighth-grade teacher from New York stated, “[My teacher education program] glossed over special ed and inclusion really quickly.” Some of the participants were required to take a special education course in either undergraduate or graduate school, but very few of these courses addressed specific instructional strategies for students with LD. Mainly, the special education courses provided an overview of special education and focused on the various laws associated with special education students. In fact, one participant from New Hampshire recalled learning in her undergraduate and graduate classes that, “When you need to modify lesson plans, you just go to the special ed teacher.” Finally, some of the participants commented that anything they learned about inclusive strategies came from “on-the-job training” and the “experience of teaching” (teachers from New York and New Hampshire). DISCUSSION The findings from the survey and follow-up telephone interviews revealed three major issues that need to be addressed as school districts attempt to provide inclusive mathematics


education to students with LD, and meet federal and state mandates for inclusion of all students in standardized testing. These issues include (a) the lack of teacher knowledge regarding the mathematical learning needs of included students, (b) the importance of teacher collaboration, and (c) the inadequacy of teacher preparation for inclusion at both the preservice and in-service levels. Issue One: Limited Understanding of Mathematical Learning Needs of Students with LD Many of the interview participants seemed to lack a strong understanding of specific pedagogical strategies to strengthen the mathematical learning of students with LD. Even though more than one-half of the survey respondents perceived themselves as quite comfortable or very comfortable in their abilities to adapt instruction for students with LD, a rather alarming percentage (between 5 percent and 13 percent) indicated that they were not comfortable meeting the mathematical learning needs of students with LD in many areas. This is problematic considering that two-thirds of the survey respondents, as well as most of the interview participants, believed that they were the primary people responsible for modifying instruction for their students with LD. Also, of great concern was that the majority of participants believed that there was no distinction between a student with LD and a low-performing student. Therefore, the participants believed that the modifications (e.g., slower pace, fewer equations per page, etc.) they used for low-performing students would be sufficient for students with LD. Many of the participants did not seem to understand that students with LD have a whole host of individualized learning challenges that need to be addressed through instructional modifications and individualized lesson plans. Although some interview participants did name constructive strategies such as chunking material, tiering lessons, and small-group work, there was no mention of other instructional strategies that research has proven effective when teaching mathematics to students with LD. For example, mathematical comprehension of students with LD can be fostered through encouraging these students to “discuss, critique, explain, and when necessary, justify their interpretations and solutions” (Cobb et al., 1991, p. 6). Students with LD can be encouraged to share their mathematical thought processes through journal writing and other forms of written expression (Thornton, Langrall, & Jones, 1998). During the interviews, none of the participants mentioned having students reflect, through words, on the process of solving mathematical equations. In addition, “written communication” was one of the learning needs for which survey respondents felt less comfortable adapting instruction. This indicates a consistency between the survey and interview data and emerges as a key issue in mathematics instruction. In addition, there was no specific mention of using individualized lesson plans, simulations, computer-assisted instruction, self-regulation strategies, or teacher modeling. Such instructional strategies have been found to clarify abstract mathematical concepts and processes by creating concrete illustrations; increase accuracy and understanding of

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higher-level mathematical concepts; strengthen estimation skills; create kinesthetic awareness; and increase independence, motivation, attention, and coordination for students with LD (Jarrett, 1999; Miller, Butler, & Lee, 1998; Steele, 2002). Although some participants mentioned that instructional resources (e.g., computer software) were available, they said that they did not really rely on such resources. Lastly, the concept of prioritizing lesson objectives for students with LD and focusing on the “big ideas” within the content is critical in mathematics instruction for students with LD, yet none of the participants raised issues related to this. Often, “mathematics programs attempt to cover exhaustive lists of learning objectives, with little or no attempt to prioritize those objectives on the basis of their relative importance later” (Carnine, 1997, p. 135). It is the teacher’s responsibility to extract the most significant material and focus on getting students to understand this information. In addition, participants did not have a sound understanding of the definition of an instructional strategy. For example, none of the participants realized that modifications such as the use of colored markers or overhead projectors are not considered instructional strategies, but are tools to enhance instruction. Many participants mentioned “differentiated instruction,” yet they were unable to define this term or give specific examples of what differentiated instruction entails. These results are consistent with the findings of other studies, which have found that general educators did not prepare written, individualized instructional plans for students with LD and did not use many of the instructional methods that researchers have proposed as effective for students with LD (deBettencourt, 1999; Schumm et al., 1995). Many researchers have argued that curricular modifications and the ways in which the curriculum is delivered are integral to creating effective mathematics programs for students with LD, and instruction should be geared toward the individual needs of students with LD (Carnine, 1998; Jones, Wilson, & Bhojwani, 1998; Montague, 1998; Rivera, 1998). The findings of the present study indicated that the majority of the participants did not modify their mathematics curriculum through prioritizing, better integration of mathematical topics, or dividing difficult topics between general education and special or resource room teachers. Issue Two: Teacher Collaboration Results indicated that the most valuable resource for general educators who taught mathematics in inclusion programs was other people—mainly special education teachers, aides, guidance counselors, and/or school psychologists. Teachers in schools with high levels of ancillary support consistently were in favor of inclusion and had higher feelings of efficacy about adapting instruction and curriculum. Many of the interview participants indicated that they met weekly or biweekly with the special education experts in their school. Whether it was advice on the ways in which to handle a specific student or simply to gain a deeper understanding of a certain disability, the participants looked to their colleagues who had special education backgrounds to provide them with assistance. Some of the general education participants even

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sought the counsel of other general educators who taught inclusion. Collaborative strategies and a genuine team mentality were the central reasons the general educators were able to endure the challenges of their mathematics inclusion classes and transform these challenges to some level of success. The results support the findings of Brownell and Pajares (1999), McLeskey and Waldron (2002), and Miller and Savage (1995), who cite collegiality as a key component in the success of inclusion programs.

tional strategies and ways to individualize lesson plans for teaching mathematics to students with LD. Consistent with the findings of Brownell and Pajares (1999), teachers with more intensive and specific professional development felt better prepared to teach in inclusion settings. Administrators must recognize the need for in-service workshops that specifically address mathematical content in areas of identified need such as using technology and teaching with the graphing calculator.

Issue Three: Inadequacy of Preservice and In-Service Teacher Preparation for Inclusion

Limitations in Design

Consistent with findings of Smith and Smith (2000) and Rao and Lim (1999), the majority of the respondents in the current study agreed that their preservice teacher preparation programs did not equip them with the necessary skills to face the challenges of teaching students with LD in mathematics inclusive classrooms. Many of the required special education courses were survey-type courses that gave an overview of special education, including broad descriptions of disabilities (mainly physical disabilities) and special education laws. The participants had similar reflections on their mathematics methods courses, which neither addressed inclusion nor discussed specific mathematics instructional strategies for students with LD. Some of the interview participants felt that the most beneficial education came from their actual experience teaching students with LD in inclusive classrooms, their on-the-job-training. Further, only a small percentage of survey respondents believed that their teacher education programs helped them develop instructional philosophies about teaching mathematics to students with LD. Therefore, teachers felt that they were sent into mathematics inclusive classrooms without being given the opportunity to reflect on their instructional beliefs and create a personalized doctrine to guide them. When teachers are faced with new challenges that they do not feel prepared for, such as teaching students with LD, having a clear set of values and instructional principles to refer to may help alleviate some of the inherent difficulties. Helping teachers develop an innovative and progressive mathematics instructional philosophy is critical, and teacher education faculty must understand that most preservice teachers address mathematics in the way in which they were taught (Parmar & Cawley, 1995). Although perceptions are changing, many preservice teachers still think of mathematics as merely rote memorization and rules. The idea of problem solving and constructivist frameworks is almost unimaginable. As one sixth-grade teacher from New York said, “Strategies are so different than what I remember math being.” As is evident from the survey results, administrative support was a significant factor in how teachers felt about inclusion and about providing effective instruction to address specific learning needs across topics. Training was sporadic or nonexistent in many cases, based on interview responses. Even those who did have some training as first-year inclusion teachers indicated that such training decreased (or was eliminated) after their first year. Frequently, professional development workshops were seen as ineffective as they did not focus on specific instruc-

The present study extends our understanding of the relationship between teachers’ beliefs, their knowledge of disability, and their self-perceived ability to provide mathematics instruction in inclusive settings. However, the current research also had its share of limitations. First, because there was no central mailing list that coincided with the required sample criteria, it was difficult to randomize the sample, as well as obtain an equal representation of respondents from all geographic regions. Second, because classroom observations were not conducted, the data reflect only teachers’ perceptions of instruction, which may be quite different from actual practice. These limitations lead to the need for caution when interpreting the study’s results. Implications for Practice The national data on percentages of students with LD receiving education in general education programs and the findings of the present study lead to some implications for practice. 1. Teachers need to broaden their repertoire of instructional and curricular modifications to better meet the needs of all students. With increasing trends toward inclusion and reductions in support, general education teachers will be taking on more and more responsibilities for included students, and they need to prepare in every way possible. The participants in this study clearly had a limited understanding of effective mathematical instructional methods for students with LD, and even expressed frustration over their perceived inability to motivate their included students. However, the majority of them firmly believed that they were responsible for teaching the students with LD in their inclusive classrooms. Yet, most of the participants never read any research articles or other material that would enhance their understanding of LD and strategies for teaching mathematics to students with LD. It is understandable that with planning, grading, attending meetings, and the everyday routine of actual instruction, a teacher’s day is quite hectic, and often there is little time left for additional work. However, general educators must find the time to advance their knowledge regarding mathematics instruction for students with LD. 2. Preservice teacher preparation programs need to be restructured to increase the amount of information provided on the learning needs of included students and pedagogical practices for diverse learners. All


mathematics undergraduate teacher education programs should require preservice teachers to spend time observing inclusive classrooms, student teaching in an inclusive classroom, and engaging in discussions of effective strategies for teaching students with LD in inclusive classrooms, particularly for challenging topics such as fractions, decimals, geometric formulas, and computer spreadsheets. It will also prepare them for working with students with attention and motivational difficulties. Using strategies such as class discussions, role playing, and journal writing, teacher education programs could also encourage teachers to reflect on their beliefs, their perceptions, and their knowledge of inclusive practice. 3. In-service teacher training that focuses on particular mathematics topics and strategies for teaching students with LD also appears to be necessary. Workshops need to address more than a few high-profile disability categories such as Attention Deficit Hyperactivity Disorder and autism, provide age-appropriate strategies for middle school–age students that are actual instructional modifications rather than just use of tools (highlighter, etc.), and provide information on how to make curricular modifications through prioritization and integration of topics. 4. Teacher collaboration should be fostered through administrative arrangements that allow for joint planning time, conference time, and sustained co-teaching experiences across several years. Teachers also need to work at developing collaborative relationships with professionals outside their immediate subject area or grade level. Colleagues are the most important source of support and information regarding effective inclusive practices.

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APPENDIX NAME: DATE: E-MAIL: STATE:

1. How many years have you been teaching mathematics? 2. How many years have you been teaching mathematics inclusion? 3. How well did your undergraduate or graduate school prepare you for teaching in an inclusive classroom? 4. Please provide some instructional strategies that you utilize for your students with LD. 5. Which specific mathematics topics do you think require instructional adaptations for students with LD? 6. Please provide specific examples of curricular adaptations you have made for your inclusion classes. 7. What resources are currently available to aid you with instructing included students? 8. What has been your greatest challenge in teaching inclusion?

About the Authors Janet R. DeSimone is Assistant Professor of Educational Leadership at Lehman College/CUNY. Her research interests are inclusion, co-teaching, and learning disabilities. Rene S. Parmar is Professor of Measurement and Evaluation and Special Education at St. John’s University. Her research interests are mathematics learning disabilities and educational assessment.