Mathematics Textbooks, Materials, and Manipulatives

Mathematics Textbooks, Materials, and Manipulatives Monica A. Lambert Florida Atlantic University Research indicates that math difficulties of students with learning disabilities (LD) begin in the elementary years and continue throughout high school (Mercer, 1992). For example, at the elementary level, students have problems with basic facts and computation that often continue through secondary school. For many students with LD, poor instruction is a primary cause of their math difficulties (Carnine, 1991; Cawley, Fitzmaurice-Hays, & Shaw 1988; Hammill & Bartel, 1995; Kelly, Gersten, & Carnine, 1990). Also, the quality of curriculum design can have a significant impact on student learning (Kelly et al., 1990). Numerous mathematics textbooks, materials, and manipulatives are available to assist teachers in providing instruction to students with LD. However, textbooks, materials, and manipulatives should be carefully evaluated before they are used in the classroom to ensure that they meet student needs. Teachers can adapt and enhance the content of traditional basals to best meet the students' needs (Kelly et al. 1990) (see Lock in this series for adaptations ideas). The purpose of this article is to describe features of (a) mathematics textbooks, (b) commerciallyprepared materials that may be used to supplement textbooks, and (c) manipulatives that are effective in teaching skills and concepts. Textbooks MATHEMATICS TEXTBOOKS This section focuses on a description of mathematics textbooks commonly found in most classrooms, a discussion of their strengths and weaknesses, considerations for using textbooks, and modifications for students with LD. Mathematics textbooks are an integral part of instruction for the classroom teacher. Typically, textbooks are the core of mathematics instruction. They consist of sequentially planned materials for teachers and students in grades K-6 (Silbert, Carnine, & Stein, 1981), and are designed for students who are learning math for the first time (Silbert et al., 1981). Textbooks are divided into units according to topics for a school year. In general, math textbooks serve as useful guides with a scope and sequence for the teacher to follow, and include goals and objectives, lesson ideas, practice exercises, and enrichment activities. Individual units focus on related skills such as numeration, addition, subtraction, multiplication, division, and so forth (Silbert et al., 1981). Textbooks serve as a guide for teaching the goals and objectives of the curriculum. Unfortunately, in many classrooms the mathematics textbook is the sole determinant of the content of daily instruction, thus, the structure of the textbook defines the curriculum. Many teachers may start at page one and continue through the text page by page, based on the assumption that following the sequence designed by experts" is best practice. Consequently, the concepts taught may be determined only by how much of the text material is covered by the end of the school year. According to Carnine (1991) teachers, on the average, spend less than 30 minutes of instructional time on some topics throughout the school year. This teaching for exposure is due to the number of topics covered in textbooks (Carnine, 1991). Teachers are trying to cover too many topics in a year; this maybe a result of the spiral curriculum that was designed to add successive depth instead of superficial coverage of topics each year (Engelmann, Carnine, &

Steely, 1991). The scope and sequence of most textbooks at the elementary level (K-S) cover number concepts, addition, subtraction, multiplication, division, decimals, fractions, time, money, and measurement. Often the teacher does not begin teaching unfamiliar concepts, such as fractions, until the tenth or twelfth week (Schmalz, 1990) as a result of following the textbook sequence. Teachers of students with LD should assess students' math levels and use this information as a guide along with the student's Individualized Education Program (IEP) to determine an appropriate scope and sequence for designing an individualized instructional program. Schmalz (1990) recommended making an outline of the topics to be covered during the year, first introducing new topics then covering topics to which students have been exposed (e.g., numeration, reading and writing numbers, basic facts in addition, subtraction, multiplication). In fact, looking at the textbook as a guide rather than a determiner of daily instruction best meets the needs of students with LD (Schmalz, 1990). Strengths and Weaknesses Because textbooks are common materials in classrooms, they should be analyzed carefully in terms of strengths and weaknesses for students with learning problems. One strength of textbooks is their comprehensiveness. They include most of the scope and sequence of elementary mathematics instruction (Silbert et al., 1981). Some textbooks also provide a course and placement guide that indicates the number of days to allow for instruction in each chapter. However, there are several weakness in textbooks. First, there is the lack of specificity (Silbert et al., 1981); that is, they do not include specific directions or strategies for how to present skills or correct student errors. Second, textbooks offer inadequate practice and review as a result of the spiral curriculum design (Silbert et al., 1981) in which each mathematical concept is expanded year after year but insufficient time is spent on concepts during each school year. The goal of the spiral curriculum is to add depth to topics each year, but due to the amount of concepts presented, the result is inadequate coverage of many topics year after year (Engelmann, Carnine, & Steely, 1991). Porter (1989) reported that a large percentage of topics in mathematics receive only brief coverage during the school year. Third, textbook designs offer a particular instructional sequence if teachers follow a chapter-by-chapter progression. According to Woodward (1991), textbooks favor massed practice (concentrated practice), lack adequate distributed practice (over time and content), and emphasize mixed practice (different skills and concepts). Inadequate practice may explain some of the confusion and inconsistent error patterns often demonstrated by students with LD. Lack of distributed practice does not allow for generalization and maintenance of skills, and mixed practice, it presented too quickly, can contribute to confusion and lack of mastery. Fourth, textbooks lack specific strategies to teach concepts and skills (Silbert et al., 1981). This is especially problematic for students with LD who must be taught a step-by-step plan for solving mathematical problems (Goldman, 1989). Engelmann et al. (1991) identifies six deficiencies based on a review of evaluation results of math textbooks: 1. Insufficient instruction in prior knowledge (e.g., definitions of key terminology). 2. Rate of concept introduction is too fast. 3. Strategies in text are too general. 4. Many instructional activities are unclear. 5. Inadequate transition from guided practice to independent practice. 6. Reviews of previously learned material is sparse or omitted. Considerations for Using Textbooks

Textbooks should be used wisely to meet the individual needs of learners. For example, the textbook can be an excellent guide and resource for developing scope and sequence based on the students' IEPs. The text can then be supplemented by manipulatives and daily review and practice of previously learned skills in order to reinforce mastery. Special worksheets also can be designed for practice instead of using the problems in the textbook. This enables the teacher to limit the number of problems and to select problems that best cover the skill learned. Teachers should also try to teach math units or themes (see Kataoka & Patton in this series), incorporating a variety of materials to meet the needs of students at all skill levels. Table 1 offers questions for teachers to consider when selecting textbooks. Modifications for Students with Learning Disabilities Textbooks may need to be modified to meet individual needs of students with LD. In this respect, it is important to keep in mind the student's instructional level and IEP goals and objectives. Students with LD must be taught specific strategies for math problem-solving. They may need (a) extra practice to reach mastery; (b) shortened assignments, as opposed to the 30 to 50 problems given in the textbook and on worksheets; and (c) maintenance practice on previously mastered skills to ensure continued mastery of the concept. The following five steps can be used to modify a math unit (Silbert et al., 1981). 1. Prioritize the objectives of the unit and set mastery levels. 2. Select problem-solving strategies. 3. Construct teaching formats for major skills and for preskills when necessary. 4. Select practice examples. 5. Design worksheets or select pages of the text to provide review of previously taught skills. MATHEMATICS MATERIALS Many mathematics materials (e.g., workbooks, kits, duplicating masters, computer software, specific skill programs, or programs for low achievers) are available for teaching mathematical skills and concepts. Mathematics materials may be used to supplement (e.g., provide additional practice or reinforcement of a skill) the general education textbook for students with LD, or they may be the main source for the lesson. When teaching to the needs of the learner, teachers should consider the student's IEP goals and objectives and the student's interest level to determine appropriate materials. This section focuses on commercially-prepared materials and includes scales to evaluate materials and computer software. Mathematics Kits Mathematics kits usually include a teacher's guide, workbooks or duplicating masters, scope and sequence charts, and in some cases, manipulatives. Most kits focus on a specific skill (e.g., basic facts, time, measurement, money). Kits are useful for introducing a lesson or unit, supplementing textbook lessons, remediation, extended practice, or enrichment activities. Some kits also can be used in a math learning center. Specific Skill Programs Specific skill programs focus on a related group of skills (e.g., addition, subtraction,

multiplication, division, fractions). Some programs are carefully planned while others are a collection of worksheets for extra practice (Silbert et al., 1981). There are strengths and weaknesses in using specific skill programs. A strength of specific skill programs is that they are likely to include adequate practice for mastery (Silbert et al., 1981). Weaknesses include lack of comprehensiveness (Silbert et al., 1981) and limited review of skills previously taught. In addition, when using specific skill programs, teachers must ensure that the program's skills match school district curricular requirements (Silbert et al., 1981). Programs for Low Achievers Programs for low achievers are designed for students who are having difficulty or who are likely to have difficulty in school (Silbert et al., 1981). Teachers should be careful when using these programs because some programs are well constructed while others are not (Silbert et al., 1981). For example, some programs jump from one skill to the next without a scope and sequence. Others do not give adequate directions, specific step-by-step strategies, or concrete examples for students to follow. Silbert et al. (1981) found that programs for the lowest performing students were the least effective. However, since the late 1970s program construction has improved. Authors of these programs have focused on presenting skills in a clearer, more concise format (Silbert et al., 1981). Silbert et al. (1981) developed an Instructional Materials Rating Scale (see Table 2) designed to assist classroom teachers in selecting appropriate instructional materials. Instructional materials are rated in four areas: strategy, sequence, examples, and practice and review. Computer Software Programs Computer software programs in mathematics can be used for teaching skills and processes (e.g., numeration, basic facts, decimals, fractions, money, measurement, geometry, algebra), drill and practice, simulations, and problem-solving. Some programs present skills in a game format, which is motivating for students, and most provide immediate feedback and the opportunity for self-correction. In addition, many programs keep a record of student performance. Teachers should view software programs before student use to ensure that the programs are appropriate for individual student needs. Directions should be clear on how to use the program and a system for monitoring student progress should be available. Table 3 provides questions teachers should consider when selecting software. MANIPULATIVES Manipulatives can be used to teach a variety of topics (e.g., place value, decimals, fractions, measurement, geometry, algebra) by providing meaningful practice opportunities and getting students actively involved in learning. Manipulative aids can help children understand and develop mental images of mathematics concepts (Dunlap & Brennan, 1979); that is, they give students a concrete basis from which abstract thinking may develop. Thus, instruction should begin with concrete experiences to assist students in understanding an abstract concept (Dunlap & Brennan, 1979), transition to semi-concrete materials, and then to abstract symbols (i.e., numbers) (Dunlap & Brennan, 1979; Mercer, 1992; Miller & Mercer, 1993). Miller and Mercer (1993) determined that students need up to seven lessons using manipulatives (concrete) and pictures (semi-concrete) before being able to transfer learning to abstract-level problems. Research indicates that the goal of mathematical curricular activities is to build stepwise associations through practice experiences so the learner does not always need manipulatives and

is able to think about the set of experiences (Miller & Mercer, 1993). As instructional planning occurs, teachers should consider levels of learning, grade level usage, and recommendations for using manipulatives. Levels of Learning Dunlap and Brennan (1979) devised a sequence of three steps that enable student to transition from concrete to abstract instructional procedures and assist teachers in determining when students are ready to move the next level. The 3 steps are listed below. Step 1 Enactive or Concrete Level The student manipulates objects to correspond with mathematics symbols. The student interprets mathematical symbols and manipulates objects to illustrate the problem. Step 2. Iconic or Semi-Concrete Level The student interprets pictures of sets and relates to mathematical symbols. The student interprets mathematical symbols and creates pictures of sets. Step 3. Symbolic or Abstract Level The student understands mathematical symbols presented alone. At the enactive or concrete level the teacher works with the student to create a mental picture of a given concept, using blocks or other counters to solve a number sentence given orally.next, the student solves number sentences with manipulative objects. At this point, textbook exercises can be solved with the use of manipulatives. Once the student develops an understanding at this level, instruction moves to the semi-concrete level where the student uses pictures of sets to solve mathematical problems. Then, the student can take mathematical sentences and draw pictures to illustrate the sets. After the student has mastered this level, he or she is ready to move to the highest level of representation, the abstract or symbolic level. Here the student can read a mathematical sentence, form a mental picture, and obtain the correct answer. If a student is having difficulty with a concept, it may be necessary to reteach--moving the student through the levels of learning again to ensure skill comprehension. Grade Level Usage Although many teachers recognize that manipulatives are appropriate for primary-level mathematics (Williams, 1986), they often fail to recognize that manipulatives are beneficial at all age levels. Research indicates that manipulative objects are effective at middle school and high school levels. Specifically, Driscoll (cited in Kennedy, 1986) stated that manipulatives not only help children in intermediate grades develop new concepts, but can also be used to provide remediation on nonmastered skills. According to Driscoll If there is any risk related to the use of manipulatives in these grades, it derives from their being ignored or abandoned too quickly (p. 7). Manipulatives are a useful tool in teaching pre-algebra and algebra. Students sometimes fail to see the big picture due to the many skills required for success in algebra, but through the use of manipulatives the teacher can teach students to understand algebraic concepts and relationships (Williams, 1986). There are also commercial materials designed to teach higher

level math concepts. For example, Creative Publications Alternative Algebra for grade 7 through college uses manipulative-based assignments to involve students actively in developing and understanding algebraic concepts. This is also a good resource for developing themes in algebra. Guidelines 1. The manipulatives must support the lesson s objectives. 2. The maniuplatives must accurately illustrate the actual mathematical process being taught. 3. More than one manipulative should be used to introduce a process. 4. The manipulative must involve moving parts or be something that is moved to illustrate a process. 5. Orient students to the manipulatives and corresponding procedures. 6. Plan for the use and fading out of manipulatives as students gain an understanding of the concept. 7. Learning comes not from the object themselves, but from the student s physical actions with the objects. 9. The manipulative must be used individually by each student. 10. A direct correlation must exist between the process illustrated by the manipulative and the process performed with pencil and paper. 11. Assist students in building self-monitoring skills (e.g., teach them how to learn) (Dunlap & Brennan, 1979, p. 90; Ross & Kutz, 1993, p. 256). These guidelines can be used by mathematics teachers in the general education classroom and by special educators who work with students with LD. It is important to keep in mind that students learn at different paces. Some may be ready to move on to the abstract level of thinking while others require concrete instruction. Manipulatives do not automatically lead to understanding; it is how they are used that is important (Baroody, 1989). Also, manipulatives must be carefully chosen to ensure the developmental appropriateness and provide a successful learning experience for the student (Baroody, 1993). In addition, teachers must avoid withdrawing manipulative aids too quickly because students with LD may have difficulty making connections to semi-concrete and abstract levels (Kelly et al., 1990). Further, when using manipulatives with students with LD it is important to maintain structured situations to assist students in organizing their thinking to allow them to see relationships or follow the computational procedure. CONCLUSION Teachers must understand math concepts and their logical teaching sequence (Mercer, 1992). A variety of textbooks, materials, and manipulatives are available for teaching mathematics. However, it is imperative that teachers examine the goals and objectives on student IEPs and use this information when selecting textbooks, materials, and manipulatives. Following the guidelines presented here and making modifications, where necessary, can assist the teacher in providing appropriate instruction. Textbooks should be used as guides, supplemented with other materials including manipulatives, when appropriate. REFERENCES Baroody, A. J. (1989). One point of view: Manipulatives don't come with guarantees.

Arithmetic Teacher, 37(2), 4-5. Baroody, A. J. (1993). Introducing number and arithmetic concepts with number sticks. Teaching Exceptional Children, 26(1), 6-11. Carnine, D. (1991). Curricular interventions for teaching order thinking to all students: Introduction to the special series. Journal of Learning Disabilities, 24, 261-269. Cawley, J. F., Fitzmaurice-Hays, A. M., & Shaw, R. A. (1988). Mathematics for the mildly handicapped. Boston: Allyn and Bacon. Dunlap, W. P., & Brennan, A. H. (1979). Developing mental images of mathematical processes. Learning Disability Quarterly 2, 89-96. Engelmann, S., Carnine, D., & Steely, D. G. (1991). Making connections in mathematics. Journal of Learning Disabilities, 24(5), 292-303. Goldman, S. R. (1989). Strategy instruction in mathematics. Learning Disability Quarterly, 12, 43-55. Hammill, D. D., & Bartel, N. R. (1990). Teaching students with learning and behavior problems (6th ed.). Austin, TX: PRO-ED. Kelly, B., Gersten, R., & Carnine, D. (1990). Student error patterns as a function of instructional design: Teaching fractions to remedial high school students and high school students with learning disabilities. Journal of Learning Disabilities, 23, 23-29. Kennedy, L. M. (1986). A rationale. Arithmetic Teacher, 33(6), 6-7. Mercer, C. D. (1992). Students with learning disabilities (4th ed.). New York: Merrill. Miller, S. P., & Mercer, C. D. (1993). Using data to learn about concrete-semiconcreteabstract instruction for students with math disabilities. Learning Disabilities Research & Practice, 8(2), 89-96. Porter, A. (1989). A curriculum out of balance: The case of elementary school mathematics. Educational Researcher, 18(5), 9-15. Ross, R., & Kurtz, R. (1993). Making manipulatives work: A strategy for success. Arithmetic Teacher, 40(5), 254-257. Schmaltz, R. (1990). The mathematics textbook: How can it serve the standards? Arithmetic Teacher, 14-16. Silbert, J., Carnine, D., & Stein, M. (1981). Organizing mathematics instruction. Columbus, OH: Merrill. Williams, D. E. (1986). Activities for algebra. Arithmetic Teacher, 33(6), 42-47. Woodward, J. (1991). Procedural knowledge in mathematics: The role of the curriculum. Journal of Learning Disabilities, 24(4), 242-250. UPDATED REFERENCES Ainsa, T. (1999). Success of using technology and manipulatives to introduce numerical problem solving skills in monolingual/bilingual early childhood classrooms. Journal of Computers in Mathematics & Science Teaching, 18, 361-369. Carnine, D., Jitendra, A. K., & Silbert, J. (1997). A descriptive analysis of mathematics curricular materials from a pedagogical perspective: A case study of fractions. Remedial and Special Education, 18, 66-81. Heuser, D. (2000). Mathematics workshop: Mathematics class becomes learner centered. Teaching Children Mathematics, 6, 288-295. Krech, B. (2000). Math: Model with manipulatives. Instructor, 109(7), 6-7. Perry, B., & Howard, P. (1997). Manipulatives in primary mathematics: Implications for learning and teaching. Australian Primary Mathematics Classroom, 2(2), 25-30. Rust, A. L. (1999). A study of the benefits of math manipulatives versus standard

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Table 1. Questions to Consider when Selecting Textbooks 1. Are lessons and activities presented in a clear, step-by-step format? 2. Does the textbook include specific directions for presenting skills? 3. Are strategies offered to correct student errors? 4. Does the pace of textbook instruction need to be modified? 5. Does the scope and sequence need to be modified to meet student needs? 6. Does the textbook require supplemental materials for additional practice? 7. Does the textbook transition from guided practice to independent practice?

Table 2. Instructional Materials Rating Scale Name of Program Year Area Examined I.Strategy Poor Excellent A.Presentation of the strategy by the teacher is carefully specified to ensure clarity and maintain consistency for related problem types. B. Presentation of the strategy is designed for systematic transition from a highly structured presentation to a less structured one. 1 1 2 2 3 3 4 4 5 5 II. Sequence A.All preskills are taught sufficiently prior to introduction of the strategy to allow for development of mastery of the preskill. B. A problem type is not introduced until students have been taught a strategy to solve problems of that type. III. Example Selection A.A mix of current and previous problem types is provided. IV. Practice and Review A.Sufficient examples are presented to enable students to master new skills. B. Sufficient review of skills is included to facilitate retention. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Stein/Silbert/Carnine, DESIGNING EFFECTIVE MATHEMATICS INSTRUCTION 3/E, 1997. Electronically reproduced by permission of Pearson Education, Inc., Upper Saddle River, New Jersey.

Table 3. Computer Software Guidelines Software Program Cost 1 Does it have a student management system? 2. Does it print, view, and delete records? 3. Does it have adjustable speeds? 4. Does it provide a range of difficulty levels? 5. Does it have sound control? 6. Does it allow teachers to add math problems? 7. Does it provide immediate feedback? 8. Does it provide individual practice? 9. What are the instructional design features?