Douglas H. Clements
Too often students with learning disabilities receive limited mathematics instruction. This is due in part to special education teachers feeling uncomfortable teaching mathematics. This leads to an overemphasis on training skills. There are three reasons for this focus on skills. First, there is a major misconception that skill learning is the bedrock of mathematics, upon which all further mathematics must be built. Second, Skills are easier to measure and teach. Third, teachers often believe that students' perceived memory deficits imply the need for constant repetition and drill.
Lessons from Research
Decades of research indicate that students can and should solve problems before they have mastered procedures or algorithms traditionally used to solve these problems (National Council of Teachers of Mathematics, 2000). If they are given opportunities to do so, their conceptual understanding and ability to transfer knowledge is increased (e.g., Carpenter, Franke, Jacobs, Fennema, & Empson, 1997).
Indeed, some of the most consistently successful of the reform curricula have been programs that
- build directly on students strategies;
- provide opportunities for both invention and practice;
- have children analyze multiple strategies;
- ask for explanations.
Research evaluations of these programs show that these curricula facilitate conceptual growth without sacrificing skills and also help students learn concepts (ideas) and skills while problem solving (Hiebert, 1999).
What is remarkable is that similar principles apply to students with learning disabilities. Many children classified as learning disabled can learn effectively with quality conceptually-oriented instruction (Parmar & Cawley, 1997). As the Principles and Standards for School Mathematics illustrates (National Council of Teachers of Mathematics, 2000), a balanced and comprehensive instruction, using the child's abilities to shore up weaknesses, provides better long-term results. For example, students may benefit less from intensive drill and practice more from help searching for, finding, and using patterns in learning the basic number combinations and arithmetic strategies (Baroody, 1996).
Many of the lessons we have learned from research for general education students apply, with modification of curse, to students with special needs as well. A particularly important one is "less is more." That is, in mathematics and science, we have found that sustained time on fewer key concepts leads to greater overall student achievement in the long run. Compared to other countries that significantly outperform us on tests, U.S. curricula do not challenge students to learn important topics in depth (National Center for Education Statistics, 1996). We state many more ideas in an average lesson, but develop fewer of them, compared to other countries (Stigler & Hiebert, 1999). Thus, U.S. students would be better off focusing on in-depth study on fewer important concepts. Such an approach is critical with students with learning disabilities. They need to concentrate on mastering the key ideas, and these ideas are not arithmetic algorithms. Even proficient adults use relationships and strategies to produce basic facts. They tend not to use traditional paper-and-pencil algorithms when computing.
Another research lesson is that a variety of instructional materials is beneficial in meeting the needs of all students. Students who use manipulatives in their mathematics classes usually outperform those who do not (Driscoll, 1983; Greabell, 1978; Raphael & Wahlstrom, 1989; Sowell, 1989; Suydam, 1986). Manipulatives can be particularly helpful to students with learning disabilities.
Somewhat surprising, manipulatives do not necessarily have to be physical objects. Computer manipulatives can provide representations that are just as personally meaningful to students. Paradoxically, computer representations may even be more manageable, flexible, and extensible than their physical counterparts (Clements & McMillen, 1996). Students who use physical and software manipulatives demonstrate a greater mathematical sophistication than do control group students who u se physical manipulatives alone (Olson, 1988). Good manipulatives are those that are meaningful to the learner, provide control and flexibility tot he learner, have characteristics that mirror, or are consistent with cognitive and mathematical structures, and assist the learner in making connections between various pieces and types of knowledge. For example, computer software can dynamically connect pictured objects, such as base ten blocks, to symbolic representations. Computer manipulatives can play those roles. They help children generalize and abstract experiences with physical manipulatives.
Recommendations for Classroom Practice
Researchers provides several recommendations for meeting the needs of all students in mathematics education.
1. Keep expectations reasonable, but not low.
Low expectations are especially problematic because students who live in poverty, student who are not native speakers of English, students with disabilities, females, and many non-white students have traditionally been far more likely than their counterparts in other demographic groups to be the victims of low expectations Expectations must be raised because "mathematics can and must be learned by all students" (NCTM, 2000). Raising standards includes increased emphasis on conducting experiments, authentic problem solving, and project-based learning (McLaughlin, Nolet, Rhim, & Henderson, 1999).
2. Patiently help students develop conceptual understanding and skills.
Students who have difficulty in mathematics may need additional resources to support and consolidate the underlying concepts and skills being learned. They benefit from the multiple experiences with models and reiteration of the linkage of models with abstract, numerical manipulations.
Expand time for mathematics. In general, the traditional curriculum does not allow adequate time for the many instructional and learning strategies necessary for the mathematics success of learning disabled students (Lerner, 1997).
Students with disabilities may also need increased time to complete assignments. Finally, they may also benefit from more time or fewer examples on tests or from the use of oral rather than written assessments.
3. Build on children's strengths
This statement often is little more than a trite pronouncement. But teachers can reinvigorate it when they make a conscientious effort to build on what children know how to do, relying on children's own strengths to address their deficits.
4. Build on children's informal strategies
Even severely learning disabled children can invent quite sophisticated counting strategies (Baroody, 1996). Informal strategies provide a starting place for developing both concepts and procedures.
5. Develop skills in a meaningful and purposeful fashion.
Practice is important, but practice at the problem solving level is preferred whenever possible. Meaningful purposeful practice gives [missing words] the price of one. Meaningless drill may actually be harmful to these children (Baroody, 1999; Swanson & Hosky, 1998).
6. Use manipulatives wisely
Manipulatives can help learning disabled students learn both concepts and skills (Mastopieri, Scruggs, & Shiah, 1991). However, students should not learn to use manipulatives in a rote manner (Clements & McMillen, 1996). Make sure students explain what they are doing and link their work with manipulatives to underlying concepts and formal skills.
7. Use technology wisely
It is important that all students have opportunities to use technology in appropriate ways so that they have access to interesting and important mathematical ideas. Access to technology must not become yet another dimension of educational inequity (NCTM, 2000). Computers can serve many purposes (Clements & Nastasi, 1992; Mastopieri et al., 1991; Pagliaro, 1998; Shaw, Durden, & Baker, 1998). Computers with voice-recognition or voice-creation software can offer teachers and peers access to the mathematical ideas and arguments developed by students with disabilities who would otherwise be unable to share their thinking. Computers can also serve as a valuable extension to traditional manipulatives that might be particularly helpful to special needs students (c.f. Weir, 1987).
Students should learn counting and arithmetic strategies but should also learn to use calculators for some purposes (Lerner, 1997). For students who can demonstrate a clear understanding of an operation, the calculator might be the primary means of computation (Parmar &Cawley, 1997).
8. Make connections
Integrate concepts and skills. Help children link symbols, verbal descriptions, and work with manipulatives. Use every possible social situation to provide meaningful situation for mathematical problem solving opportunities. (Baroody, 1999; Parmar & Cawley, 1997).
9. Adjust instructional formats to individual learning styles or specific learning needs
Formats might include modeling, demonstration, and feedback; guiding and teaching strategies; mnemonic strategies for learning number combinations; and peer mediation (Gersten, 1985; Lerner, 1997; Mastropieri et al., 1991). Use projects and games to help the teacher guide learning, rather than relying solely on "telling" (Baroody, 1999). The traditional sequence of direct teacher explanations, strategy instruction, relevant practice, and feedback and reinforcement is often effective, but the potential of students to learn through problem solving should not be ignored. Too often, direct instruction approaches squeeze out other possibilities. Use direct instruction only when students are unable to invent their own strategies. In all cases, help them make strategies explicit (Kame'enui & Carmine, 1998).
10. Emphasize statistics, geometry, and measurement as well as arithmetic topics (Parmar & Cawley, 1997)
All students need access to varied topics in mathematics. Topics beyond arithmetic are increasingly important in our day-to-day lives.
Overall, solve problems, encourage reasoning, and use modeling. With patience and support, these processes are also in the reach of most children.
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Time to prepare this material was partially provided by two National Science Foundation (NSF) Research Grants, ESI-9730804, "Building Blocks—Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development" and ESI-9814218: "Planning for Professional Development in Pre-School Mathematics: Meeting the Challenge of Standards 2000." Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the NSF.
Douglas H. Clements, Professor of Mathematics, Early Childhood, and Computer Education at SUNY/Buffalo, has conducted research and published widely in the areas of the learning and teaching of gemoetry, computer applications and the effects of social interactions on learning. He has co-directed several NSF projects, producing Logo Geometry, Investigations in Number, Data, and Space, and more than 70 referred research articles. Active in the NCTM, he is editor and author of the NCTM Addenda materials and was an author of NCTM's Principles and Standards for School Mathematics (2000). He was chair of the Editorial Panel of NCTM's research journal, the Journal for Research in Mathematics Education. In his current NSF-funded project, Building Blocks-Foundations for Mathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development, he and Julie Sarama are developing mathematics software and activities for young children.
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