Instruction for Elementary Students with Learning Disabilities.

ABSTRACT: Recent research in mathematics instruction requires educators to rethink long-established beliefs about teaching, learning, and assessment. In particular, this research underscores the need for problem solving and higher level thinking in mathematics. Consistent with these recommendations, this article presents and illustrates four promising themes for mathematics instruction that have emerged from research involving students with learning disabilities. These themes--(a) providing a broad and balanced mathematics curriculum; (b) engaging students in rich, meaningful problem tasks; (c) accommodating the diverse ways in which children learn; and (d) encouraging students to discuss and justify their problem-solving strategies and solutions--suggest ways for rethinking the teaching and learning of mathematics in relation to students with learning disabilities. The curriculum, assessment, and professional teaching Standards of the National Council of Teachers of Mathematics (NCTM; 1989, 1991, 1995) call for strategic shifts in mathematics instruction for all students. In essence, these shifts involve a movement toward higher level mathematical reasoning and problem solving, and involve rethinking long-established beliefs about teaching, learning, and curricular practices. Common practice in both general and special education classrooms, however, still reflects a narrow emphasis on computation. This focus is also mirrored in diagnostic teaching and evaluation thrusts (Heshusius, 1991). Not only are such perspectives on instruction and assessment incompatible with the vision of the Standards, but they are also contrary to the findings of recent research about mathematics teaching and learning (e.g., Carpenter, Fennema, Peterson, Chiang, & Loef, 1989; Englert, Tarrant, & Mariage, 1992; Resnick, 1987, 1989; Thornton & Bley, 1994). The National Council of Teachers of Mathematics (1989) proposed five goals for rethinking mathematics teaching and learning.

The council held that students should (a) learn to value mathematics, (b) become confident in their ability to do mathematics, (c) become mathematical problem solvers, (d) learn to communicate mathematically, and (e) learn to reason mathematically. To accomplish these goals, the council advocates, teachers should decrease their emphasis on complex paper-and-pencil computation, rote memorization of rules and formulas, written practice, "one answer, one method," and teaching by telling. These recommendations for school mathematics are grounded in constructivist theory (e.g., Cobb & Bauersfeld, 1995; Noddings, 1990) and stem from a broad research base in mathematics education (Grouws, 1992). Little of this research, however, has focused specifically on mathematics instruction for students with learning disabilities (LD). Furthermore, few studies involving students with learning disabilities have focused on higher level mathematical thinking and problem solving (Parmar & Cawley this issue; Marshall, 1988; Mastropieri, Scruggs, & Shiah, 1991). A major goal of the present article was to describe four major themes related to higher level thinking and problem solving that have emerged from recent mathematics studies involving students with learning disabilities.

These themes are as follows: * Providing a broad and balanced mathematics curriculum; * Engaging students in rich, meaningful problem tasks; * Accommodating the diverse ways in which children learn; and * Encouraging students to discuss and justify their problem-solving strategies and solutions. In essence, these themes embrace the philosophy that students with learning disabilities benefit from rich, challenging programs that promote mathematical thinking. Each of these themes will be illustrated by case study examples drawn from one or more of four research reports. Prior to considering the case studies, we present an overview of each in the next section. These overviews are intended to provide a context for describing and discussing the case studies.

STUDIES INVOLVING STUDENTS WITH LD

STUDY 1: CHILDREN'S UNDERSTANDING OF MULTIDIGIT NUMBERS This study, by Jones et al. (1996), validated a framework that described and predicted various levels of young children's thinking related to multidigit number sense. The framework itself was used to generate and evaluate two different versions of an instructional program emphasizing number sense in general education classrooms that included students with identified learning disabilities. The instructional approach based on the framework, grounded in social constructivism (e.g., Cobb & Bauersfeld, 1995), took the position that students' opportunities to construct mathematical knowledge arise from attempts to resolve conflicting points of view in a group, from attempts to reconstruct and verbalize a mathematical idea or solution and, more generally, from attempts to reach consensus with others.

The instructional program was consistent with the recommendations of Englert et al. (1992) and Heshusius (1991), who suggested that students with disabilities should be challenged with meaningful problem tasks that promote multiple solutions and strategies. Throughout the study, all students were encouraged and given time to work collaboratively to solve problems at whatever levels they could attain. A further expectation was that all students would share and justify their thinking in different ways. Differences in the understandings demonstrated by children in the two instructional groups were attributed largely to the quality of the problem-solving experiences and the level of student interactions.

STUDY 2: THE USE OF REFLECTIVE ANALYSES ON THE INSTRUCTIONAL PRACTICES OF PROSPECTIVE ELEMENTARY TEACHERS. This research, by Langrall, Thornton, Jones, and Malone (1996), used a case study approach to investigate the effects of reflection on prospective elementary teachers' instructional practices in mathematics. Recommendations from the NCTM (1989, 1991) Standards documents provided the basis for the teachers' reflections on their instructional experiences in elementary classrooms. Six prospective teachers participated in a series of 11 instructional classroom experiences. In the first and last of these sessions, the teachers instructed small groups of students, including children with learning disabilities. Video analysis and transcripts of these lessons were supplemented with other data, including semistructured interviews, stimulated-recall sessions, written documents submitted by the teachers, and researcher field notes. These data sources became the context for six case studies that documented the changes in instructional strategies adopted by the teachers. This intervention resulted in powerful changes in the instructional practices of these prospective teachers, including changes in the way they related to students with learning disabilities.

Major changes included (a) greater use of problem-driven tasks and open-ended questions, (b) increased expectations for student reasoning and multiple-solution strategies, (c) greater emphasis on student dialogue and collaboration; and (d) less teacher-directed instruction. Although the focus of this study was on prospective teachers, the research also captured rich vignettes of student interactions with each other and with their teacher.

STUDY 3: SUPPORTING MIDDLE SCHOOL STUDENTS WITH LD IN THE MAINSTREAM MATHEMATICS CLASSROOM The major goal of this research project, by Borasi, Packman, and Woodward (1991), was to institute a comprehensive professional development program that would encourage and support middle school mathematics teachers in rethinking their teaching goals and practices so as to better respond to the learning needs of all their students, with special attention to those with learning disabilities.

The project was developed by an interdisciplinary team comprising mathematics educators, an expert in learning disabilities, and both mathematics and special education teachers. Its theoretical framework was characterized by a constructivist perspective on knowledge and learning, an information-processing model of learning disabilities, and an "inquiry approach" to teaching mathematics. At the core of this project was the development of three units intended to illustrate an "inaction" inquiry approach in middle school classrooms that included students with learning disabilities. Extensive case study data on the students with LD were collected during the implementation of these units in different instructional settings (a private school for students with learning disabilities, and both urban and suburban public school general education classrooms). The results suggest that an inquiry approach, complemented by appropriate instructional modifications and adaptations, can help mathematics teachers meet the challenges that accompany a diverse student population. Furthermore, in mathematics classrooms informed by such an approach, students' learning differences could be capitalized upon and turned into assets in the learning setting.

STUDY 4: MATHEMATICAL PROBLEM-SOLVING PROCESSES OF PRIMARY-GRADE STUDENTS IDENTIFIED WITH LD Behrend's (1994) study examined the problem-solving processes of five second- and third-grade students identified as learning disabled. In accordance with the recommendations of their Individualized Education Programs (IEPs), these students received daily mathematics instruction in an LD resource room. Cognitively Guided Instruction (Fennema & Carpenter, 1985) provided the framework for assessing the children's independent and assisted problem-solving abilities.

Data were gathered during individual interviews and small-group sessions. During the group sessions, students were presented with word problems, given time to solve them, and encouraged to share their strategies in whole-group discussions. Prompts, from general to more explicit, were given only when needed. Behrend (1994) found that, given the opportunity, the students in her study were capable of sharing their strategies, listening to other children's strategies, discussing similarities and differences among strategies, justifying their thinking, and helping each other understand word problems. Although students modeled their solutions for each other, she observed that teacher modeling of solution strategies was rarely needed and generally did not promote better problem solving among the children. All five students could solve a variety of problems, including difficult addition, subtraction, multiplication, and division word problems; problems with extraneous numerical information; and multiple-step problems. Behrend (1994) found that the students in her study were capable of generating and utilizing their own problem-solving strategies and did not need to be taught specific strategies. Based on this finding, she questioned the need for explicit strategy instruction in mathematics for students with learning disabilities and recommended instructional approaches that utilize students' available problem-solving processes.

CASE ILLUSTRATIONS OF THE FOUR THEMES Four case illustrations from the studies described above exemplify the themes presented in this article. Although some of these cases could illustrate more than one theme, the discussion highlights salient features of each theme in turn.

PROVIDING A BROAD AND BALANCED MATHEMATICS CURRICULUM Overview of Theme. Baroody and Hume (1991) noted that many children who experience learning difficulties in mathematics, including those who have learning disabilities, are "curriculum disabled." For such students, Trafton and Claus (1994) recommended a broader and more balanced curriculum, in contrast to a more traditional curriculum, with its repetitive and unnecessary emphasis on computation. A broader curriculum can be established by utilizing problem-driven instruction that incorporates a greater emphasis on number sense and estimation, data analysis, spatial sense and geometric thinking, patterns and relationships leading to algebraic understandings, and the supportive use of technology (National Council of Teachers of Mathematics, 1989). Broadening the curriculum to include a variety of mathematical domains does not preclude, but actually encourages, the development of appropriate mathematical skills. In essence, it presents opportunities for different kinds of thinking and success beyond numerical reasoning. Such a curriculum should be reflected in children's IEPs. Although it does not entail disregarding computation in the IEP, a broadened thrust enables students with learning disabilities to use mathematics more flexibly, insightfully, and productively (Bley & Thornton, 1994; Borasi, in press; Englert et al., 1992).

Case Illustration. To exemplify the "broad and balanced curriculum" theme, this section draws on case data from two students with identified learning disabilities. These illustrations include an episode involving a student we will call Jana, focusing on mental math (Jones et al., 1996), and an episode documenting "Terrell's" thinking about a geometry task (Langrall et al., 1996). On the Wechsler Intelligence Scale for Children-III, 9-year-old Jana scored well below average on the Verbal Comprehension and Mathematics subtests. Because she also had severe receptive language and auditory memory difficulties, she was placed in a self-contained learning disabilities program. Despite her difficulties with word problems, her good visual memory and interest in mathematics led to her being jointly placed in a general classroom for mathematics, where the following episode occurred.

The activity had started the moment Mrs. Tate's class entered the room. Each pair of children selected a card that showed an amount of money they could spend. The task was to "purchase" items from the garage sale mural on the wall, spending as much of their money as possible. The children worked in pairs for a short time before Mrs. Tate brought them together to share their thinking. Over the past 2 weeks the garage sale activity had constituted a daily math problem for the children--one that had grown out of earlier problem-solving experiences with addition and money. Jana spoke for herself and her partner: "We picked the picture frame for 38¢ and the poster for 15¢--and we just have 7¢ change." When asked to explain how they knew they would have 7¢ change, Jana said, "We just thought about the 100s Chart. We started with 38 and went down to 48, then counted 5 more. So we paid 53¢ -- that gives us 7¢ back because we had 60¢ to spend."

This episode illustrates how the 100s Chart enabled Jana to move beyond paper-and-pencil computation. During early instruction with this graphic aid, Jana was encouraged to move a finger along the chart as she counted. She then grew able to visualize the counting-on process just by thinking of the 100s Chart. In this case the chart was an appropriate compensatory tool that enabled Jana to compute two-digit sums mentally.

The second case centers on Terrell, a student with learning disabilities enrolled in a general fifth-grade class. He showed little ability to reason abstractly and had visual-perception problems, but retained information once he had internalized it. The following episode illustrates how Terrell, his learning difficulties notwithstanding, manipulated pattern blocks and drew on his understanding of a high-dive flip to reason about angle measures in a meaningful way. Terrell pointed to the three trapezoids he had arranged around a dot on the overhead [see Figure 1].

He explained what Duane had told his group about how a "360 flip" off the high dive "goes all the way around." "Here three of these [trapezoids] go all the way around. So we divided 360 by 3 and got 120 for the big angle." As he observed groups at work, the teacher, Mr. Adams, had not been sure that Terrell understood Duane's explanation of a 360-degree turn, so he was pleased when he heard Terrell rephrase the explanation to Duane and later volunteer to present the group's solution. Mr. Adams's expectation that all group members would be able to present the group's solution set the stage for Terrell to verbalize his solution strategy within his working group. Expectations like this that encompass opportunities for children with learning disabilities to articulate their thinking have been shown to help with learning and retention (see Montague, this issue). Other groups in the class had found different ways of showing that the obtuse angle of the pattern block was 120 degrees. The initial task had challenged each group of four students to determine the measures of each of the angles of the pattern blocks. As part of the ensuing discussion, a summary chart was made to organize the class' findings. These case illustrations document how two students with learning disabilities were successful in mathematics programs that emphasized a broad, balanced curriculum. When instruction is consistent with the tenets of such a curriculum and different approaches are valued, it is possible for children to achieve success within their specific limitations (Bulgren & Montague, 1989; Cawley, Fitzmaurice-Hayes, & Shaw, 1988; Ginsburg, this series).

ENGAGING STUDENTS IN RICH, MEANINGFUL PROBLEM TASKS Overview of Theme. Recent recommendations (e.g., National Council of Teachers of Mathematics, 1989, 1991; National Research Council, 1990) highlight the need for relevant, problem-driven instruction. A central thesis of these recommendations is that all students should become confident "doers" of mathematics and, consequently, be capable and resourceful problem solvers. This requires that all students have the opportunity to explore many different types of mathematical problems and that they be both expected and encouraged to use a variety of strategies in solving them (National Council of Teachers of Mathematics, 1989).

Although problem solving traditionally has been a difficult area for many students with learning disabilities (Montague & Bos, 1986; Wansart, 1990), Bulgren and Montague (1989) reported that these students can succeed beyond current expectations if they are exposed to developmentally appropriate, meaningful problem tasks that are complemented by appropriate instructional modifications. Moreover, children experiencing difficulties with formal computation or basic fact recall should not be prohibited from engaging in more challenging problem-solving tasks (Cawley & Miller, 1989; Ginsburg, this series).

In fact, a substantial body of research highlights the effectiveness of using problem solving as the vehicle for learning mathematics, including basic facts and computation (e.g., Carpenter & Moser, 1984). When problem tasks are sufficiently complex, rich, and open-ended, they can be explored at different levels of understanding. Stenmark (1991) characterized a "rich" problem in three ways: (a) The problem leads to other problems, (b) the problem raises other questions, and (c) the problem has many solution approaches. We would add a fourth criterion: that a problem makes multiple connections. Case Illustration. One example of a complex, rich problem task, which was presented to a self-contained class of nine students classified as severely learning disabled, is the following triangle--rectangle problem: Is every triangle 1/2 of a rectangle? Yes or no? Prove it.

The following excerpt from the teacher's journal provides insight into both the nature of the problem task and the thinking and physical representations the students used to solve it. Three boys, working together, cut out the colored triangles [from the Triangle Worksheet (TWS; see Figure 2)], taped them to the triangles on the white TWS [See Figure 3], and formed parallelograms. Their premise was, "No--two equal triangles do not form rectangles. The shapes formed are not rectangles because they do not have 90 degree angles." (Stone, 1993, p. 54) A [second] group of two boys gave themselves permission to cut the ... triangles [see Figure 3] on the altitude and tape the two [colored] triangles, one on either side, to the white triangle. They had a little difficulty with #3, the obtuse triangle. They cut off a piece along a line at the end. After they taped the two pieces to the existing triangle they had a small piece sticking out on the left and a small hole on the right. They asked if it was O.K. to cut off the piece and move it. They ended up with a perfect rectangle with a base of 4 and a height of 3 (very ingenious!). (Stone, 1993, p. 56)

One girl was working by herself due to absenteeism. She also cut the cut-out triangles. She worked totally independently. Her first conjecture was that all triangles except #3 could form rectangles. She was very proud of herself when she finally figured out how to do #3 like the other boys did. (Stone, 1993, p. 54) Within problem contexts such as this, students with learning disabilities are able to draw upon their diverse strengths as they solve problems using different parameters and achieve success "within their specific limitations" (Borasi, in press).

This kind of exploration addresses the broader need to challenge students to think beyond common expectations. When viewed according to the characteristics proposed by Stenmark (1991), the triangle--rectangle problem meets the criteria for a rich problem in that it (a) generated extension problems, (b) raised questions about shapes, (c) generated different solutions by redefining the parameters of the problem, and (d) set the stage for exploring further connections. In relation to extension problems, the group of children in the first scenario just mentioned correctly reasoned that every triangle is not half a rectangle but is half a parallelogram. This raised a further problem that was pursued in a later lesson, "Is every triangle half a parallelogram?"

The second group of three boys redefined the problem in their own way and, in essence, investigated an extension problem: "Can a rectangle be formed by physically changing two congruent triangles?" The problem raised questions about the defining properties of shapes. For example, when is a parallelogram a rectangle? The problem also provided an opportunity for the teacher to follow up on the distinction between congruent shapes and shapes that have the same area. Because of differing interpretations and reasoning, the problem gave rise to two different but valid solutions. In one case, assuming that the shape of the triangle could not be changed, the children concluded that it was not possible for every triangle to be half a rectangle. In the other case, the children made a different assumption--that the shape of the triangle could be changed as long as areas remained the same. In this situation it was possible to construct a rectangle that was twice the area of the given triangle. In terms of connections, the triangle--rectangle problem task set the stage for the teacher to make the link between areas of triangles and areas of rectangles. The natural connection between the visualization of length--width measures of a rectangle and the corresponding base--height measures of a triangle could be highlighted in such instruction. Further, it would be possible to make connections among the areas of a triangle, a rectangle, and a parallelogram. When students are given ongoing opportunities to engage in rich problem tasks, as in this case, the results can be quite dramatic. This success is consistent with research documenting the fact that students learn what they have an opportunity to practice. Students who have had many opportunities to solve mathematical problems become better at problem solving (e.g., Carpenter et al., 1989; National Council of Teachers of Mathematics, 1989; Silver, 1985).

ACCOMMODATING THE DIVERSE WAYS IN WHICH CHILDREN LEARN Overview of Theme. Mathematics today is viewed as a "sense-making experience" involving numerical, logical, and spatial concepts and relationships. Because sense-making is idiosyncratic, students with learning disabilities generally need considerable time to understand problem situations and construct strategies.

Further, if these students are to develop higher levels of mathematical thinking and more positive dispositions toward mathematics, they need ongoing opportunities to explore mathematical tasks in ways that match their learning strengths (Speer & Brahier, 1994). For example, learning groups might be formed on the basis of complementary learning styles. With this approach, students with different strengths can find their niches and achieve success within their specific limitations (Borasi, in press).

Case Illustration. An episode involving Dan (a pseudonym; Behrend, 1994) is a case in point. Dan, a 9-year-old, received mathematics instruction in a learning disabilities resource room. On the Wechsler Intelligence Scale for Children--Revised, his Full Scale IQ score was average, although he had difficulty processing multiple pieces of information. At the time of the study he was on medication to control his attention-deficit disorder. Dan was the most inconsistent student with respect to mathematical performance in Behrend's (1994) study. His inconsistency was readily apparent in routine computational problems, where he attempted to apply learned rules in a nonmeaningful way. For example, when asked which of two ways (see Figure 4a) would be the better way to find the total, Dan selected the example on the left because it corresponded to his interpretation of the teacher's rule for adding: "The ones are first" (p. 74). Dan believed that 78 was a reasonable answer because the 4 was "where it's supposed to be.... Because that's where you mostly put the first number for numbers" (p. 75). However, when Dan was faced with a nonroutine problem and was allowed the flexibility to solve it in his own way, he demonstrated surprising insight, as illustrated by his solution to the following problem. 19 children are taking a bus to the zoo. They will have to sit either two or three in a seat. The bus has 7 seats. How many children will have to sit three to a seat, and how many can sit two to a seat? (Behrend, 1994, p. 77) Dan quickly drew seven lines to represent seats, drew a circle for each seat, and repeated the process until he had accounted for 19 circles (see Figure 4b). This kind of modeling and counting strategy exemplifies Dan's thinking in problem situations for which a known procedure was not readily available. Not only was he capable of solving nonroutine problems like this, but Behrend reported that he was also capable of correctly solving problems that included extraneous information. In effect, when the teacher accommodated Dan's distinctive learning style, Dan was successful; when he felt compelled to use the teacher's algorithmic approach, he invariably failed. The inflexibility of a meaningless procedure appeared to inhibit his ability to recognize the reasonableness of an answer or his attempts at alternative strategies. As this case illustrates, accommodating the diverse ways in which children learn does not always require proactive strategies on the part of the teacher. Rather, there are times when the teacher needs to step back and observe and listen to children's thinking patterns so that he or she can respond to and maximize the children's strengths. In her study, Behrend (1994) found that students with learning disabilities constructed and utilized their own strategies to solve a wide variety of problem types. She concluded that instruction should build on children's current understandings and promote the development of increasingly more efficient problem-solving strategies, rather than emphasizing specific rules and procedures. In finding implications for instruction, Behrend generates a powerful message for teachers of diverse learners: A model of instruction which involves posing problems, allowing students time to solve the problems in their own way, listening to students' strategies, assisting only when necessary, and discussing similarities and differences between strategies provides several advantages over the other forms of instruction. Teachers are able to make assessment an integral part of instruction, students are given more control over their learning, and mathematics is seen as a process of making sense of number relationships. Instruction becomes less a matter of following directions, or imitating what has been modeled, and more a way of making connections to what is already known. (p. 109)

ENCOURAGING STUDENTS TO DISCUSS AND JUSTIFY THEIR PROBLEM-SOLVING STRATEGIES AND SOLUTIONS Overview of Theme. Research has suggested that classrooms in which students "discuss, critique, explain, and when necessary, justify their interpretations and solutions" (Cobb et al., 1991, p. 6) are effective at nurturing mathematical thinking. Such inquiry-oriented approaches are consonant with Scheid's (1990) review of research in special education, which has also emphasized the importance of children's thinking about their solutions and justifying them. Students can communicate and justify their thinking and reasoning through journal writing, partner sharing, or whole-class discussion, depending on the situation and individual student needs. After completing a problem task, teachers could invite students to share their thinking or journal entry with a partner or small group. In this way, all students have an opportunity to communicate their thinking in some way, whether or not they subsequently share their ideas with the larger group.

This Think-Pair-Share approach (McTighe & Lyman, 1988) increases the kinds of personal communications that are necessary for students to internally process, organize, and retain ideas (Pimm, 1987). Whole-class discussions in which students explain and justify their solutions to problems provide a rich forum in which students develop their understanding of mathematics. In sharing their ideas, students take ownership of their learning and negotiate meanings, rather than relying solely on the teacher's authority (Cobb et al., 1991). Lo, Wheatley, and Smith (1991) also reported positive changes in students' dispositions and self-esteem when they were expected to listen to each other and respect others' ideas. Students with diverse learning needs gain credence with their peers by reporting to the whole class what they have learned from participating cooperatively in group work or journal writing. Reporting sessions also provide opportunities for less articulate students to learn from their peers who, in a sense, serve as role models for higher level thinking. Repeated exposure to such experiences enhances the likelihood that students with specific disabilities will begin to independently think at higher levels (Scheid, 1990).

Case Illustration. Borasi, Kort, Leonard, and Stone (1993), reporting on a class of nine children with severe learning disabilities, noted how these students frequently wrote to explain to others what they did, and then paired up for sharing. In fact, the two children who also had attention-deficit/hyperactivity disorder were often invited to share their thinking while walking down the hall so as to "get rid of some excess energy" (p. 143).

In another example from Borasi et al. (1993), students were asked to write a journal entry describing their processes for finding the number of tiles needed to tessellate the classroom floor. One student, whom we will call Todd, had a severe motor disability in writing as well as a "numerical" disability. He was helped by Borasi, a participant observer in the class, to first reconstruct and then record his solution in a journal. Prompted by the researcher's questions, Todd explained how he had solved the tiling problem. He declined her offer to scribe for him, preferring to do it himself. He described each step of his solution process aloud before writing it down. The journalizing task was completed over a 2-day period, with support and questioning from Borasi. In the end Todd produced a well-organized and understandable entry, which Borasi later transcribed on a computer for sharing with other students. Reflecting on Todd's experience, the researcher commented that this one on one work seemed really important and productive with a student with severe writing disabilities such as [Todd]; it was our hope that this experience would show [Todd] what he could really do, and provide a model for the future; we do not expect him now to be able to do similar writing on his own yet, but perhaps he might be able to do it a second time around with less help, and gradually learn to do the same without the adult support. (Borasi et al., 1993, p. 152)

CONCLUDING COMMENTS Consistent with recent recommendations of the National Council of Teachers of Mathematics (1989, 1991), this article has presented and illustrated four promising themes for mathematics instruction that have emerged from recent mathematics studies involving students with LD.

These themes--(a) providing a broad and balanced mathematics curriculum; (b) engaging students in rich, meaningful problem tasks; (c) accommodating the diverse ways in which children learn; and (d) encouraging students to discuss and justify their problem-solving strategies and solutions--suggest ways for rethinking the teaching and learning of mathematics for students with learning disabilities. Case data exemplifying these themes provide a vision of what can happen when teachers nurture mathematical thinking and provide time and opportunity for students to engage in and share their solutions to rich, meaningful problems. Students with cognitive and processing disabilities deserve--and have the potential--to be empowered mathematically. In the field of learning disabilities, relatively few studies reflect the instructional themes identified in this article. This article has highlighted four studies that illustrate successful engagement of students with LD in problem solving and higher level thinking. Consistent with the findings of these studies, we are recommending a broadened approach to curriculum and instruction that accommodates and capitalizes on diversity in thinking and learning. Although further research is needed, the studies outlined in this article suggest that the mathematical abilities of students with LD can be accommodated and capitalized upon when these students have pervasive opportunities to learn in challenging, broad, and well-balanced programs. The fact that students may need appropriate compensatory techniques notwithstanding, our thesis is that programs based on the themes presented in this article can raise the mathematical thinking of these students to levels previously considered to be beyond their reach. If you don't let your grasp extend your reach, then you'll never extend your reach. --Woody Allen, 1992

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AUTHOR: Carol A. Thornton, Cynthia W. Langrall, and Graham A. Jones

Carol A. Thornton, PhD, is a distinguished university professor in the Department of Mathematics at Illinois State University. She has conducted extensive research on children's mathematics learning in the context of classroom instruction, has special interests in curriculum and staff development involving both general and special education teachers, and is the co-author of Teaching Mathematics to Students with Learning Disabilities (published by PRO-ED) and the coeditor of a special education book for the National Council of Teachers of Mathematics.

Cynthia W. Langrall, PhD, is an assistant professor in the Department of Mathematics at Illinois State University, where her research focuses on the teaching and learning of mathematics in the elementary and middle grades, with a special emphasis on instructional practices that accommodate student diversity.

Graham A. Jones, PhD, is a visiting professor in the Department of Mathematics at Illinois State University. Dr. Jones's interests include research on children's mathematical thinking in number, probability, and problem solving. He has also conducted and reviewed research in teacher education and staff development that have an impact on both general and special education. Address: Carol A. Thornton, Illinois State University, 4520 Mathematics Department, Normal, IL 61790-4520.

SOURCE: Journal of Learning Disabilities 30 142-50 Mr/Ap '97. Reproduced with permission from Journal of Learning Disabilities, copyright 1997 by the National Council of Teachers of Mathematics. All rights reserved.