Inclusion of Students with Special Needs in the

Abstract

The National Council of Teachers of Mathematics recently published Principles and Standards for School Mathematics: Discussion Draft (1998b). Included in the Equity Principle is a definition for "all students". Upon review of this definition, as well as the draft itself, students with special needs seem to have been overlooked. With the recent trend of including students with disabilities in the regular education programs, certain modifications for implementing the curriculum will need to be made for these students to be successful.

The purpose of this paper is to suggest to the authors of the Standards to specifically mention students with special needs in the definition and also provide suggestions of strategies that aid in the instruction of these students in the regular education classroom. Recommendations where these strategies can be addressed in the Principles and Standards for School Mathematics are mentioned as well.

A major emphasis in the National Council of Teachers of Mathematics (NCTM) Standards (1991, 1995) and a guiding principle in the Principles and Standards for School Mathematics: Discussion Draft (Standards 2000, NCTM, 1998b) is the commitment to the learning of mathematics by all students. The Equity Principle in the Standards 2000 draft (1998b) restates the definition of "all students", first included in the Professional Teaching Standards (NCTM, 1991). The authors defined "all students" as:

- students who have been denied access in any way to educational opportunities as

well as those who have not;

- students who are African American, Hispanic, American Indian, and other minorities

as well as those who are considered to be part of the majority;

- students who are female as well as those who are male;

- students who have not been successful as well as those who have been successful in

school and in mathematics (NCTM, 1991, p. 21-22).

The NCTM member handbook (NCTM, 1998a) has expanded and refined this description of every child. The Board of Directors state they are concerned about students who have not been provided with education opportunities for reasons such as language, gender, physical impairment, socioeconomic status, ethnicity, and so on (NCTM, 1998a). Therefore, they reiterate that "every child" includes:

- learners of English as a second language and speakers of English as a first

language;

- members of underrepresented ethnic groups and members of well-represented

groups;

- students who are physically challenged and those who are not;

- females and males;

- students who live in poverty and those who do not;

- students who have not been successful and those who have been successful in school

and in mathematics (NCTM, 1998a, p.11).

In both definitions there is one group that is not specifically mentioned, students with special needs. This includes students with a disability and those identified as talented and gifted. Based on the recent trend of including students with special needs in the regular education program, the definition should include this broad category of students.

The purpose of this paper is to provide an explanation of why this group of students should be included in the Principles and Standards for School Mathematics (1998b). First discussed will be the rationale for specifically addressing students with special needs in the definition of "every child". Other topics discussed are the relationship between inclusive education and the NCTM Standards, the need to incorporate strategies proven to be successful for students with special needs in the Standards, and finally, suggestions of where these strategies can be included in the Principles and Standards for School Mathematics (NCTM, 1998b).

Expanding the Definition of "Every Child"

In the definition of "every child", the authors mention those students who have not been successful in school and in mathematics, those who are physically challenged, and students who speak English as a second language (NCTM, 1998a). With the specific mention of these individuals, not addressing students with special needs seems almost discriminatory. Although "students who have not been successful in school and mathematics" (NCTM, 1998a, p. 11) could be interpreted to include some students with special needs, it does not adequately represent all students with special needs. For example, students with a learning disability can be successful in the regular education class with modifications and/or accommodations. Also, students identified as talented and gifted, although they may be considered to be successful, need modifications to their program as well. Modifications are essential to the success of some students with special needs and if they are not provided, students should not be penalized for not being successful. A disability does not make a person unsuccessful. This is why students with special needs should not be implied in this part of the definition, but mentioned specifically instead. Also, how these students can be assisted in being successful in the mathematics classroom, should be expanded on in the Equity Principle as well.

Inclusive Education and the Standards 2000

In order to adequately understand the relationship between inclusive education and the Standards 2000, a description of inclusion is necessary. The best way to define inclusion is to illustrate what inclusion encompasses. In this section, inclusive settings are described in relation to the impact on students with and without disabilities and the instructional techniques implemented in these settings. Finally, the relationship between inclusion and the Standards 2000 will be examined.

Full inclusion refers to educating all students in the regular education classroom setting. This includes providing special education services to students with special needs within this setting. The focus of inclusive classes is to individualize education to meet the needs of each student.

Inclusion affects all students in the regular education class, not just the student with a disability. In inclusive classes, everyone's gifts and talents, including those traditionally defined as having profound disabilities or chronically disruptive behaviors, are recognized, encouraged, and utilized to the fullest extent possible (Stainback, Stainback, & Jackson, 1992). The sense of community within the classroom helps to foster self-esteem, pride in accomplishments, mutual respect for each other, and a sense of belonging and self-worth (Stainback, Stainback, & Jackson, 1992). Inclusive classes create the support systems that allow students to feel accepted and comfortable in their environment.

Each student's individual differences can influence their instructional needs, which calls for individualized instruction for all students. Professors Stainback (1984) state:

In short, there are not- as implied by a dual system- two distinctly different types of

students, that is, those who are special and those who are regular. Rather, all students

are unique individuals, each with his/her own set of physical, intellectual, and

psychological characteristics (p. 103).

The curriculum is adjusted to meet the needs of the students for they are not expected to achieve a predefined curriculum that does not take into account their diverse characteristics and needs (Stainback, Stainback, & Jackson, 1992).

In summary, inclusion has been shown to benefit all students, not just those with a disability. Snell (1991) identified several positive effects that inclusion has on all students. They include (a) the development of social skills, (b) the improvements in attitudes that students without disabilities have for their peers with disabilities, and (c) the development of friendships and positive relationships between students as a result of integration.

Teaching in an inclusive classroom requires modifying, not changing, the basic instructional processes to fit individual characteristics and needs. These basic instructional processes include: developing behavioral objectives, curricular-based assessment procedures, task analysis, the arrangement of antecedents and consequences, and open education/discovery learning methods (Stainback & Stainback, 1984). Gardner (1977) explains "There are no unique methods for use with exceptional children that differ in kind from those used with normal children" (p. 74).

The focus is different in inclusive classrooms- the role of the teacher is the facilitator, therefore requiring students to be actively engaged in the learning process. Responsibility is delegated to the group members for learning and supporting each other (Stainback, Stainback, & Jackson, 1992). When a child is an active participant in structuring the environment, his/her development is enhanced (Lipsky & Gartner, 1992). With the teacher as the facilitator, students are able to promote cooperation among each other by engaging in activities such as: cooperative learning, peer tutoring, buddy systems, and circle of friends (Lipsky & Gartner, 1992; Stainback, Stainback, & Jackson, 1992). This results in cooperation and collaboration with peers rather than competition, not only for the present, but for the future as well. Activities such as involving students in preparing material, sharing information with other students, acting as tutors, and monitoring their own performance have been found to offer both affective and cognitive benefits (Lipsky & Gartner, 1992).

The importance of implementing a variety of teaching methods, such as cooperative learning and peer tutoring, is documented in research. Pomplun (1997) states that "cooperative groups have been proposed as one way to increase the mainstreaming and inclusion of students with disabilities" (p. 51). Studies show that one to one and small group instruction are more effective approaches than whole group instruction for students with disabilities in regards to levels of engaged behavior (Logan, Bakeman, & Keefe, 1997). In a study by Pomplun (1997), results suggest that open-ended tasks should be used for maximum participation in cooperative groups. Reviews of literature on cooperative learning also show positive student outcomes in three areas: academic achievement, personal/social development, and interpersonal attraction (Maheady, Harper, & Mallette, 1991). In regards to another instructional technique, peer tutoring, results show that tutor and tutee make academic, as well as social, improvements (Maheady, Harper, & Mallette, 1991).

As previously stated, inclusive classrooms are student centered and allow for many cooperative learning activities. The curriculum has changed in the sense that more hands-on learning, problem solving, and issues relevant to students' lives are prevalent in the classroom. In order to meet the needs of a student with a disability who requires outside assistance, the classroom curriculum is modified to benefit not only this student, but other students as well. The objective is to have students get their educational and related needs met within the regular education classroom (Stainback, Stainback, & Jackson, 1992). This is another way in which the curriculum is fitted to the student and not the student fitted to the curriculum.

Also, the strategies that have been shown to improve the performance of students with a disability in mathematics are strategies that all students can benefit from learning. These strategies and how they fit into the NCTM Standards will be addressed in detail in succeeding sections of this paper.

Many of the same ideas or concepts that describe inclusive classes are also listed in the Standards 2000 as effective classes or what should be required in mathematics classes. For example, inclusion advocates argue that high expectations should be established for students with special needs, recognizing however, that modifications may need to be made to help the student reach these expectations. Similarly, the Standards 2000 refers to raising expectations for all students and providing support for students (NCTM, 1998b). The authors suggest one method that can be used as a support service is peer tutoring (NCTM, 1998b). This was highlighted as a technique that has shown to help students with special needs.

Another way students with special needs can be assisted in the regular education class which was not mentioned previously, is by the use of technology in the classroom. Students with a disability and their nondisabled peers should be provided with the opportunity to use technology in the classroom. This is an important accommodation for students with special needs because these resources assist them in participating in class. For example, a student who has great difficulty with short term memory can be allowed to use a calculator for solving algorithms in the classroom. This allows the student to focus on the concept of the problem instead of wasting needless time on rote mathematics. A call for access to technology was included in the Equity Principle in the Standards draft and could very easily be expanded to emphasize the importance of technology for students with special needs.

In the Teaching Principle, the authors refer to worthwhile tasks that should be implemented in the mathematics classroom (NCTM, 1998b). As previously stated, open-ended tasks can increase participation of students with special needs in the regular education setting (Pomplun, 1997) and should be encouraged. The Teaching Principle can incorporate this research into the existing discussion on good tasks in order to address ways to help students with special needs succeed.

Similar to inclusive classes, the Standards 2000 also recommends that students are actively engaged in their learning (NCTM, 1998b). This can be done through the use of manipulatives, technology, and by working together to solve problems (NCTM, 1998b). These are modifications that have shown to benefit students with special needs. The Learning Principle discusses how students should be expected to build on prior knowledge and make connections in mathematics. Since students with special needs at times lack basic skills needed to build on, the Standards 2000 should address how this impacts the implementation of the Standards.

As illustrated, many of the same concepts that are important for good inclusive classes also apply to the recommendations made in the Principles and Standards for School Mathematics. However, many opportunities in the draft are available for the integration of how this relates to students with special needs.

Strategies for Students with Special Needs

Many different strategies exist that aid in the learning of mathematics for students with special needs. Effective strategies vary for each individual child however. One strategy that works for a student with a disability, may not necessarily be as effective for another. Since the list of strategies that could be implemented is exhaustive, two major strategies highlighted in research will be discussed. Also, these two intervention techniques are either already included in the Standards 2000 or can easily be assimilated into the draft. The two intervention techniques are strategy instruction and the use of manipulatives.

In order to understand the importance of strategy instruction in the educational planning of students with special needs, a definition and description of how strategy instruction assists students with learning disabilities will be provided. Recommendations for how strategy instruction can be included in the Standards 2000 will also be given.

Process-based instructional models like strategy instruction, can assist students with learning disabilities (Montague, Applegate, & Marquard, 1993) when problem solving. These students are deficient in strategies that help students decide what to do when solving a problem. Students with a learning disability frequently do not process information or apply knowledge efficiently; lack the necessary domain-specific or general problem-solving knowledge and self-regulation processes; and have trouble selecting and deploying task-appropriate strategies (Montague, 1992). Research has shown that for students with a learning disability, using strategies when problem solving can help students organize their information, decide what to do, and therefore result in improved performance.

Strategies are typically defined as "cognitive operations employed to facilitate performance" (Pressley, Harris, and Marks, 1992). Strategy instruction can involve self-instruction, self-questioning, self-monitoring, self-regulation, and self-assessment (Case, Harris, & Graham, 1992; Montague, 1992 ). These components help students to gain access to strategic knowledge, guide execution of strategies, regulate use of strategies, monitor the problem solving process, evaluate decisions and solutions, and check computations and procedures (Montague, 1988, 1992). Strategies that use mnemonics, including acronyms or verbal rehearsal, can help students with a learning disability gain access to procedural knowledge, which has been shown as something that poor problem solvers lack (Montague, 1992). Strategies that use mnemonics, including acronyms or verbal rehearsal, can help students with a learning disability gain access to procedural knowledge, which has been shown as something that poor problem solvers lack (Montague, 1992). These mnemonics are tools that students can use to remember steps or procedures. For example, the mnemonic WORDS can help students recall steps for solving a word problem. The steps for this are: (a) read the Word problem, (b) Organize the information, (c) Recognize the Operation, (d) Draw a picture, and (e) Solve the word problem. Strategy instruction should also build in ways for students to maintain and generalize strategies taught.

In the preceding statements made regarding inclusive classes, it was implied that more students with disabilities are being included in the regular education class setting. A concern regarding full inclusion is the possibility that students with special needs will not be full participants in the learning process. It is believed that strategy instruction could increase students' self-confidence, self-esteem and motivation, and therefore result in more frequent participation in the inclusive class (Montague, Applegate, & Marquard, 1993). Also, if strategies involve techniques for maintenance and generalization, students may utilize these strategies in other settings and situations.

Although direct teaching is a component of strategy instruction and the Standards 2000 encourage students to be actively engaged in the learning process, strategy instruction can still be incorporated. After student exploration and cooperative learning activities, students who have difficulty retaining steps or procedures, can be taught a strategy to help them remember. These students can be active participants by helping to create a mnemonic that is meaningful to them. Also the standards 2000 emphasizes the need to have prior knowledge of concepts to expand learning of mathematics further. For students who have difficulty retrieving prior knowledge, strategies can provide that link and make recall easier.

By making students actively engaged in their learning process, material relevant to their lives, and moving from concrete to abstract learning, students will hopefully gain a more positive attitude towards mathematics. The student with a learning disability in particular has been shown to benefit from a carefully structured, active approach to learning mathematics (Bley & Thornton, 1995). Grover, Hojnacki, Paulson, and Matern (1994) state that student understanding and performance results increase when abstract concepts are tied to concrete models, or manipulatives.

Zental and Ferkis (1993) note that students with learning disabilities performed better on tasks requiring literal use of numbers and worse on tasks which require applying mathematical concepts. One way manipulatives improve performance results is by helping students develop a concrete language for talking about mathematical concepts, thus helping students to become more at ease with the concepts themselves (Grover et al., 1994). Concrete objects used for mathematical relations may help students develop more accurate and complete mental representations than students who are not learning concretely (Marzola, 1987). Manipulatives may help children better understand mathematical ideas and applications to real life situations and can also greatly increase on-task behavior and attention span (Marzola, 1987). Students keep on task because they have ownership of ideas, are active in their learning, and are physically involved (Grover et al., 1994).

When using manipulatives, it is imperative to ensure that the bridge be made between the concrete and abstract stages (Bohan & Shawaker, 1994), especially for students with a disability. Quite often these students cannot make this connection on their own and therefore the teacher should assist in the connection of the manipulatives to the symbolic representation (Grover et al., 1994). Keeping in mind that the Standards 2000 indicate that teachers should act as a facilitator, prompts can be given by the teacher instead of a direct explanation to connect the concrete to the abstract. From the prompts, students can explore in their cooperative groups how to connect the two concepts.

Manipulatives are included in the Standards 2000 under the Representation Standard. Similar to the reasoning for why students with disabilities use manipulatives (that is, to better understand mathematics concepts by learning concretely), the Standards states that children use representations for the same reason. Representations include physical objects, such as manipulatives, but encompass more than that especially in the older grades.

Where to Include Students with Special Needs

With so many of the same characteristics shared by inclusive classes and effective classes as seen by the Standards 2000, many places where mention of students with special

needs can be included. The first two obvious places should be in the Equity and the Learning Principles.

In the Equity Principle (NCTM, 1998b), students with special needs should be included in the definition of "all students". Since this section already states the need for higher expectations, the expectations for students with special needs can be addressed here as well. Also in this principle is the mention of how technology can assist in achieving equity. As illustrated before some students with special needs require the use of technology as modifications. Mention of these modifications appears to fit well with the statements made regarding technology.

In regards to the benefits of using manipulatives, the Standards 2000 (NCTM, 1998b) should address in the Representation Standard that research supports that students with special needs using concrete objects. The other possible place to mention this is in the discussion of active engagement in the Learning Principle.

Strategy instruction can be included in the Learning Principle as well. The Standards state " ... teachers may initially encourage their students to develop their own strategies for solving problems in ways that make sense to them" (NCTM, 1998b, p. 35). The use of strategy instruction, mentioning the use of mnemonics, could be added here to illustrate how students with special needs can create or utilize strategies to assist them.

Conclusion

Under the Individuals with Disabilities Education Act, the law requires that students with special needs be placed in their least restrictive environment. With more inclusive classes established in the schools, the standards set need to include these students. Accommodations and modifications will need to be made for these students to ensure their success in the regular education setting.

The Principles and Standards for School Mathematics (NCTM, 1998b) does not appear to adequately illustrate how these standards will modified, if at all, for students with special needs. Therefore, my recommendation to the authors is to address these students in the standards, not only in the definition of "all students", but by providing explanations as to their role in the regular education mathematics class.

Two intervention techniques that have been shown to help students with special needs were described in this paper. These, or other researched techniques shown to assist students with special needs, should be incorporated into the Standards 2000 draft. To assume that these students can achieve the same standards without the appropriate modifications is unfair to the students. Therefore, students with special needs should included in The Principles and Standards for School Mathematics (NCTM, 1998b); they should not be discriminated against.

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