The following is a literature review completed in Fall 2011 for a graduate course on mathematical problem solving.
Mathematics reform is changing what we expect from students in a mathematics classroom. Teachers are moving away from the so-called “drill and kill” where the focus is on memorization and repeated practice of standard algorithms and procedures. Conceptual understanding is emphasized rather than memorization (Cole and Wasburn-Moses, 2010, p. 15). Instead of predictable, repetitive end-of-chapter problems, students are being given complex problems to solve and encouraged to develop and invent their own strategies and algorithms (Sayeski and Paulsen, 2010, p.14).
The teacher’s role, in a reform classroom, shifts from being a source of correct answers to a facilitator who engages students in mathematical conversations, where “differing mathematical ideas are shared and valued” (Cole and Wasburn-Moses, 2010, p. 16). Students must “explain their mathematical reasoning to others and follow the explanations of their peers” (Baxter, Woodward, and Olson, 2001, p. 530). Students have to come up with solutions and justify them to other students, while the teacher watches and guides the process.
Guiding the development of these new expectations has been a shift in the understanding of how people learn. Theorists have moved away from the reinforcement-based behavioral approaches and towards constructivist approaches to learning. By constructivist, we mean that students construct their own unique understanding of what’s being taught (Woodward & Montague, 2002, p.90). Two main types of constructivism, Woodward and Montague (2002) explained, are individual constructivism, which focuses on what is going on in an individual student’s head and social constructivism, which focuses on the process by which students come to understand material together (p. 90). They cautioned that people mistakenly think direct instruction and constructivism are incompatible, when in fact constructivism does not prescribe any particular teaching techniques (p. 92).
Along with these significant changes to the way that mathematics is being taught, a new emphasis is being placed on special education students accessing the general education curriculum. In many states, special education students take the same standards-based, criterion-referenced tests that general education students take. Special education students, as a matter of equity and social justice, are being expected to master the general education curriculum– which means, for mathematics, a reform curriculum. Sadly, both special educators and general educators have been shortchanging students with disabilities in their mathematics education.
General educators are not trained in the techniques needed to teach students with disabilities, and special educators are not trained on implementing mathematics reform. As Cole and Wasburn-Moses (2010) lamented, “… many general and special education teachers are being taught completely differently, at a time when collaboration is more important than ever before” (p. 14). With more and more students with disabilities being educated in general education (Fuchs, et al., 2004, p. 440), whether in Inclusion, RSP, or RTI settings, general education and special education teachers need to eb on the same page. Woodward and Montague (2002) contended that “special educators must reconsider traditional approaches and provide instruction that is more consistent with the reform agenda” (p. 91). Not all the blame can be placed on special educators, however. Mathematics reform advocates have not stepped up to the challenge of working with special educators; the landmark 1989 NCTM Standards, for example, had absolutely no mention of students with disabilities (Woodward & Montague, 2002, p. 89). Reform curricula are often designed without attention to the needs of special education students; for example, the need for additional practice in working with numeric operations and algorithms. Overall structural supports are lacking as well; special educators “report a lack of materials, a lack of support, … [and] a lack of confidence in teaching mathematics” (Cole and Wasburn-Moses, 2002, p. 14).
General and special educators clearly have not stepped up to the plate to implement a reform approach for special education students. However, can students with learning disabilities actually handle the demands and challenges of a reform approach. These changes in the way mathematics is taught present particular challenges for these students. These students “typically have deficits in attention, memory, background knowledge, vocabulary, language processes, strategy knowledge, visual-spatial processing, and self-regulation” (Jitendra and Star, 2011, p.13). These deficits lead to struggling with “generalization, applying metacognitive strategies, discriminating key points from irrelevant information, and solving multistep problems” (Cole and Wasburn-Moses, 2010, p. 15). Teaching a reform-based curriculum to students with learning disabilities, therefore, requires different techniques and methods than many educators have been using.
Three central skills to succeeding in a reform classroom are the ability to participate in class discussions, to work in small groups, and to engage in metacognitive activities. Baxter, Woodward, and Olson (2001) observed five classrooms and found that low-achieving students rarely spoke or gave only one or two-word answers (p. 8). They were generally off-task and their cognitive challenges kept them from following the discussion and deciphering what other students were saying. Baxter, Woodward, and Olson were not surprised by this, as even the classroom teachers often had trouble understanding student explanations.
Baxter, Woodward, and Olson (2001) also found that students with disabilities often had challenges in working in small groups. Although they participated more fully than in large class discussions, they often relied on the more high-achieving students for the complex mathematical thinking and were reduced to performing menial tasks such as handing manipulatives to their partner (p. 11). To help mitigate this problem, Sayeski and Paulsen (2010) recommended peer tutoring where the more skilled student coaches the student with disabilities and then they trade roles and the student with disabilities coaches the more skilled student. Training students on how to be a tutor is essential for this kind of model to succeed (Cole and Wasburn-Moses, 2010,p. 18). An alternative model that Sayeski and Paulsen (2010) commented on featured homogeneous groups composed only of students with disabilities with additional support from the teacher.
Metacognition, many reformers suggest, is “the central feature of problem solving” (Woodward and Montague, 2002, p. 96) and a “critical variable” in small group problem solving (Baxter, Woodward, Olsen, 2001, p. 544). Students with learning disabilities have challenges both in mathematical metacognition (being able to monitor and guide their own progress on problems) and behavioral metacognition (being able to monitor their own behavior and stay on task). Cole and Wasburn-Moses (2002) suggested the use of checklists with strategies that students could follow while solving problems (p. 17). Sayeski and Paulsen (2010), in a similar vein, recommended the use of a “laminated bookmark” with problem-solving prompts (p. 18).
Woodward and Montague (2002) suggested that part of these metacognitive challenges come from difficulties in problem representations, which involves “graphically representing word problems using relational schematics (p. 96). This was less of a problem in non-reform curricula, as problems were usually grouped in the textbook by solution strategy (Jitendra and Star, 2011, p. 13). So students would be taught a skill and then all of the following problems utilized the same skill. Schema-based instruction goes hand-in-hand with reform curricula by giving students with disabilities the conceptual tools needed to understand change in value relationships, part-whole relationships, and relationships between sets (Jitendra and Star, 2011, p. 14-15) rather than simply looking at key words in problems as was classically taught. The goal is to teach students to identify problems that share a common solution strategy so that students will recognize problems that are similar to problems they have already solved (Fuchs, et. al, 2004, p. 419). Students can be given the opportunity to explore and create their own schemas for problems, and then the schemas that they develop can be made explicit and then practiced in order to help students with disabilities get additional practice using different schemas in their problem solving (Cole and Wasburn-Moses, 2010, p.17). The goal is to help students to develop broad, generalized schemas that can be used to solve a wide variety of patterns (Fuchs, et al., 2004, p. 420).
Another issue that students with learning disabilities have in a reform classroom has to do with the spiral design of the curriculum. Sayeski and Paulsen (2010) observed that in reform curricula that topics occur frequently throughout the curriculum and “are not taught to mastery” (p. 14). In a more traditional curriculum, students might spend a few months on addition until they’ve mastered it, and then move onto subtraction. In a reform curriculum, however, a topic such as addition is introduced, and then the class moved onto another topic and comes back to the original topic later. In addition to spiraling, reform curricula simply move too fast for many students with disabilities (p.19). Sayeksi and Paulsen recommended that teachers respond to these concerns by creating additional problems for students to practice strategies on and by creating opportunities for intensive skill work around number sense (p. 19).
Instructional design is of particular importance, too, when teaching mathematics to students with disabilities. It is important to include “structured review of previously learned information” and to “connect the end of one lesson to the beginning of another” (Cole and Wasburn-Moses, 2010, p.17). Sayeski and Paulsen (2010) also recommended amplifying key mathematical concepts that arise during class discussion and documenting learning on the board that occurs during a lesson as “students with math LD characteristically have difficulty identify salient information and concepts from oral information” (p. 17). This kind of support will help students with learning disabilities, who otherwise miss important concepts while trying to follow the complex interplay between students in classroom discussions.
Another aspect of instructional design is in the teaching of algorithms. Traditionally special educators have taught the standard (American) algorithms and then used drill-and-practice until students mastered them. Reform curricula generally have students generate their own methods for solution and then present a range of alternative algorithms (Sayeski and Paulsen, 2010, p. 15). The worry with special education students is that they might not be able to “extract a reliable method” from the range of options (p. 16). Sayeski and Paulsen (2010) recommended identifying a single algorithm for each problem type and then teaching it “in exclusion of the other… options” (p. 16). Woodward and Montague (2002) suggested using algorithms that “make the conceptual aspects of the operation and the role of place value more explicit” (p. 95). Sayeski and Paulsen (2010) concurred about choosing algorithms that “strengthen conceptual understanding” (p. 16) and added that they need to be efficient and effective for students with learning disabilities and that many of the algorithms taught in reform curricula do not satisfy these criteria (p. 16). Balancing the reform idea of helping students to generate and evaluate algorithms must be balanced with the need for students with learning disabilities to be able to have a reliable method that they understand and can use to solve problems.
Mathematics reformers and special educators seeking to utilize the principles of reform in their teaching face a daunting task. The bar has been raised for all students, and the burden is particularly affecting students with learning disabilities. Their cognitive challenges make generating their own solution strategies, solving complex problems, participating in a community of mathematical learnings, participating in mathematical discussions in small and large-group formats, and the other expectations of a reform curriculum and pedagogy difficult. But it is a challenge we must rise to if we are wanting all students to reap the benefits of mathematics reform.
Baxter, J.A., Woodward, J., and D. Olson. (2001). Effects of Reform-Based Mathematics Instruction on Low Achievers in Five Third Grade Classrooms. The Elementary School Journal, 101(5), 529-547.
Bottege, et. al. (2007). Integrating Reform-Oriented Math Instruction in Special Education Settings. Learning Disabilities Research & Practice, 22(2), 96-109.
Cole, J. E. and L. Washburn-Moses. (2010). Going Beyond “The Math Wars.” Teaching Exceptional Children, 42(4), 14-20.
Fuchs, L.S. Et al. (2004). Expanding Schema-Based Transfer Instruction to Help Third Graders Solve Real-Life Mathematical Problems. American Educational Research Journal, 41(2), 419-445.
Jitendra, A. (2002). Teaching Students Math Problem-Solving Through Graphic Representations. Teaching Exceptional Children, 34(4), 34-38.
Jitendra, A.K. & J. R. Star (2011): Meeting the Needs of Students With
Learning Disabilities in Inclusive Mathematics Classrooms: The Role of Schema-Based Instruction on
Mathematical Problem-Solving, Theory Into Practice, 50:1, 12-19
Sayeski, K. L. and K.J. Paulsen (2010). Mathematics Reform Curricula and Special Education: Identifying Intersections and Implications for Practice. Intervention in School and Clinic 46(1), 13-21.
Woodward, J. and M. Montague (2002). Meeting the challenge of mathematics reform for students with LD. The Journal of Special Education, 36(2), 89-101.